From dd24662ab9683fe690e7f3babc1b1c20821ed886 Mon Sep 17 00:00:00 2001 From: "weiye.wang" Date: Mon, 2 Jan 2023 17:56:54 +0800 Subject: [PATCH] =?UTF-8?q?20230102=20=E5=A2=9E=E5=8A=A0=E4=BA=86=E4=B8=80?= =?UTF-8?q?=E4=BA=9B=E9=A2=98=E7=9B=AE=E7=9A=84tag?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 题库0.3/Problems.json | 5020 ++++++++++++++++++++++++++++++----------- 1 file changed, 3647 insertions(+), 1373 deletions(-) diff --git a/题库0.3/Problems.json b/题库0.3/Problems.json index 7b735586..c14f975b 100644 --- a/题库0.3/Problems.json +++ b/题库0.3/Problems.json @@ -322502,13 +322502,15 @@ "id": "020404", "content": "幂函数$y=x^{\\frac 52}$的定义域的区间表示为\\blank{50}, 幂函数$y=x^{-\\frac 13}$的定义域的区间表示为\\blank{50}, 幂函数$y=x^{-\\frac 12}$的定义域的区间表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322521,13 +322523,15 @@ "id": "020405", "content": "若函数$y=(m^2-2 m-2) x^{m+1}$是幂函数, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322540,13 +322544,15 @@ "id": "020406", "content": "已知幂函数$y=x^a$的图像经过点$(2, \\dfrac{\\sqrt 2}2)$, 则当$x=4$时的函数值为\\bracket{20}.\n\\fourch{$16$}{$\\dfrac 1{16}$}{$\\dfrac 12$}{$2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322559,13 +322565,15 @@ "id": "020407", "content": "下列幂函数中, 定义域为$\\{x | x>0\\}$的是\\bracket{20}.\n\\fourch{$y=x^{\\frac 23}$}{$y=x^{\\frac 32}$}{$y=x^{-\\frac 23}$}{$y=x^{-\\frac 32}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322578,13 +322586,15 @@ "id": "020408", "content": "幂函数$y=x^n(n \\in \\mathbf{Z})$的图像一定不经过\\bracket{20}.\n\\fourch{第一象限}{第二象限}{第三象限}{第四象限}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322597,13 +322607,15 @@ "id": "020409", "content": "求下列函数的定义域, 并作出它们的大致图像:\\\\\n(1) $y=x^{\\frac 15}$;\\\\\n(2) $y=x^{\\frac 43}$;\\\\\n(3) $y=x^{-\\frac 34}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322616,13 +322628,15 @@ "id": "020410", "content": "若幂函数$y=x^{-m^2+2 m+3}$($m$为整数)的定义域是$\\mathbf{R}$, 求$m$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322635,13 +322649,15 @@ "id": "020411", "content": "若函数$y=(m x^2+m x+2)^{\\frac 34}$的定义域是$\\mathbf{R}$, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322654,13 +322670,15 @@ "id": "020412", "content": "已知集$C=\\{f(x) | f(x)$是定义域上的单调增函数或是单调减函数$\\}$, 集$D=\\{f(x) |$在$f(x)$的定义域内存在区间$[a, b]$, 使得$f(x)$在区间$[a,b]$上的值域为$[k a,k b]\\}$($k$为常数).\\\\\n(1) 当$k=\\dfrac 12$时, 判断函数$f(x)=\\sqrt x$是否属于集合$C \\cap D$, 若是, 这求出区间$[a, b]$; 若不是, 请说明理由;\\\\\n(2) 当$k=\\dfrac 13$时, 若函数$f(x)=\\sqrt x+t \\in C \\cap D$, 求实数$t$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322673,13 +322691,15 @@ "id": "020413", "content": "当$n>0$时, 幂函数$y=x^n$的图像都通过\\blank{50}两点, 且在区间$(0,+\\infty)$上随着$x$的增大而\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322692,13 +322712,15 @@ "id": "020414", "content": "当$n<0$时, 幂函数$y=x^n$的图像都通过\\blank{50}这一点, 且在区间$(0,+\\infty)$上随着$x$的增大而\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322711,13 +322733,15 @@ "id": "020415", "content": "已知函数$y=x^a$的图像关于$y$轴对称, 且在区间$(0,+\\infty)$上为严格减函数, 其中$a \\in\\{-1,1,-2,2, \\dfrac 12, 3, \\dfrac 13\\}$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322730,13 +322754,15 @@ "id": "020416", "content": "已知实数$a, b$满足$a+b \\neq 0$, 函数$y=\\dfrac{b-x}{a+x}$的图像的对称中心是$(-1,-1)$, 则实数$a$与$b$满足的条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322749,13 +322775,15 @@ "id": "020417", "content": "下图中, $C_1, C_2, C_3$为幂函数$y=x^a$在第一象限的图像, 则函数表达式中的指数$a$依次可以取\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 3:{pow(3,-0.5)}] plot (\\x,{pow(\\x,-2)}) node [right] {$C_1$};\n\\draw [domain = 0:3] plot ({pow(\\x,3/4)},\\x) node [right] {$C_2$};\n\\draw [domain = 0:3] plot (\\x,{pow(\\x,3/4)}) node [below] {$C_3$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\dfrac 43,-2, \\dfrac 34$}{$-2, \\dfrac 34, \\dfrac 43$}{$-2, \\dfrac 43, \\dfrac 34$}{$\\dfrac 34, \\dfrac 43,-2$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322768,13 +322796,15 @@ "id": "020418", "content": "当$x \\in(1,+\\infty)$时, 下列函数的图像全在直线$y=x$下方且关于$y$轴对称的是\\bracket{20}.\n\\fourch{$y=x^{\\frac 12}$}{$y=x^{-2}$}{$y=x^2$}{$y=x^{-1}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322787,13 +322817,15 @@ "id": "020419", "content": "已知函数$y=\\dfrac x{x-1}$, 则下列四个命题错误的是\\bracket{20}.\n\\onech{该函数的图像关于点$(1,1)$对称}{该函数的图像关于直线$y=2-x$对称}{该函数在定义域内是严格减函数}{将该函数图像向左平移一个单位, 再向下平移一个单位后与函数$y=\\dfrac 1x$的图像重合}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322806,13 +322838,15 @@ "id": "020420", "content": "利用幂函数的性质比较下列两个值的大小:\\\\\n(1) $(-0.2)^{\\frac 25}$\\blank{50}$(-0.3)^{\\frac 25}$;\\\\\n(2) $(-2)^{-\\frac 25}$\\blank{50}$3^{-\\frac 25}$;\\\\\n(3) $1.68^{0.8}$\\blank{50}$(-1.71)^{0.8}$;\\\\\n(4) $0.3^{-\\frac 12}$\\blank{50}$0.2^{-\\frac 12}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322825,13 +322859,15 @@ "id": "020421", "content": "作出下列函数的大致图像:\\\\\n(1) $y=(x-1)^{-\\frac 23}$;\\\\\n(2) $y=1+\\sqrt {3-x}$;\\\n(3) $y=\\dfrac{2 x-3}{x-3}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322844,13 +322880,15 @@ "id": "020422", "content": "幂函数$y=x^{m^2-2 m-3}$($m \\in \\mathbf{Z}$)的图像关于$y$轴对称, 且与$x$轴、$y$轴均无交点. 求此幂函数的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322863,13 +322901,15 @@ "id": "020423", "content": "已知$x_1, x_2, x_3 \\in \\mathbf{R}$, 且$x_1+x_2>0, x_2+x_3>0, x_1+x_3>0$, 则$x_1^3+x_2^3+x_3^3$\\blank{50}$0$.(填大小关系符号)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322882,13 +322922,15 @@ "id": "020424", "content": "已知$a$为正实数, 函数$f(x)=-\\dfrac{2^x-1}{2^x+a}$的图像关于原点对称, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322901,13 +322943,15 @@ "id": "020425", "content": "由下列不等式, 分别求出实数$a$的取值范围.\\\\\n(1) $(a+2)^{\\frac 23}>(1-2 a)^{\\frac 23}$;\\\\\n(2) $(a+2)^{\\frac 13}>(3-2 a)^{\\frac 13}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322920,13 +322964,15 @@ "id": "020426", "content": "已知函数$y=\\dfrac{a x+1}{x+2}$(常数$a \\in \\mathbf{Z}$). 问: 是否存在整数$a$, 使该函数在$[-1,+\\infty)$上是严格增函数, 并且函数值恒不为负? 若存在, 求出所有符合条件的$a$; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322939,13 +322985,15 @@ "id": "020427", "content": "下列函数是指数函数的序号为\\blank{50}, 是幂函数的序号为\\blank{50}.\\\\\n\\textcircled{1} $y=x$; \\textcircled{2} $y=x^3$; \\textcircled{3} $y=\\mathrm{e}^x$; \\textcircled{4} $y=\\sqrt[3]x$; \\textcircled{5} $y=2^{-x}$; \\textcircled{6} $y=2^x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322958,13 +323006,15 @@ "id": "020428", "content": "下列函数是指数函数的序号为\\blank{50}.(请填入全部正确序号)\\\\\n\\textcircled{1} $y=(-2)^x$; \\textcircled{2} $y=3 \\cdot 2^x$; \\textcircled{3} $y=\\pi^x$; \\textcircled{4} $y=x^3$; \\textcircled{5} $y=2^{-x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322977,13 +323027,15 @@ "id": "020429", "content": "写出下列函数的定义域:\\\\\n(1) $y=3^{1-2 x}$:\\blank{50};\\\\\n(2) $y=0.5^{\\frac 1{2-3 x}}$:\\blank{50};\\\\\n(3) $y=\\dfrac 1{5^{\\frac x{x-1}}-1}$:\\blank{50};\\\\\n(4) $y=\\sqrt {3^{x^2-2 x}-1}$:\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -322996,13 +323048,15 @@ "id": "020430", "content": "已知常数$a>0$且$a \\neq 1$. 若无论$a$取何值, 函数$y=a^{x+2}+2$的图像恒经过一个定点, 则此定点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323015,13 +323069,15 @@ "id": "020431", "content": "函数$y=2^x$的图像与函数$y=2^{-x}$的图像关于\\blank{50}对称, 它们的交点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323034,13 +323090,15 @@ "id": "020432", "content": "在同一坐标系中分别作出下列函数的大致图像:\\\\\n(1) $y=3^x$;\n(2) $y=(\\dfrac 13)^x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323053,13 +323111,15 @@ "id": "020433", "content": "已知指数函数图像的一部分如下图所示, 求该指数函数的表达式.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.3]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,9) node [left] {$y$};\n\\draw (0,0) node [below] {$O$};\n\\draw [domain = -2:3] plot (\\x,{pow(3,-\\x)});\n\\draw [dashed] (-1,0) -- (-1,3) -- (0,3);\n\\draw (-1,0) node [below] {$-1$} (0,3) node [right] {$3$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323072,13 +323132,15 @@ "id": "020434", "content": "在平面直角坐标系中作出函数$y=-4^x$的大致图像.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323091,13 +323153,15 @@ "id": "020435", "content": "已知常数$a>0$且$a \\neq 1$. 若无论$a$取何值, 函数$y=a^{x+2}+2$的图像恒经过一个定点, 则此定点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323110,13 +323174,15 @@ "id": "020436", "content": "若指数函数$y=(m-1)^x$在$\\mathbf{R}$上是严格减函数, 则实数$m$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323129,13 +323195,15 @@ "id": "020437", "content": "已知$a>0$且$a \\neq 1$. 若$m>n$, 且$a^m1$}{$a<1$}{$a(a-1)<0$}{$a(a-1)>0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323186,13 +323258,15 @@ "id": "020440", "content": "若函数$y=2^x-m$的图像不经过第二象限, 则实数$m$的取值范围是\\bracket{20}.\n\\fourch{$m \\geq 1$}{$m<1$}{$m>-1$}{$m \\le -1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323205,13 +323279,15 @@ "id": "020441", "content": "若指数函数\\textcircled{1} $y=a^x$, \\textcircled{2} $y=b^x$, \\textcircled{3} $y=c^x$, \\textcircled{4} $y=d^x$在同一坐标系内的图像如图所示, 则$a$、$b$、$c$、$d$的大小顺序是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-0) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:2] plot (\\x,{pow(3,-\\x)}) (-1,3) node [above] {\\textcircled{1}};\n\\draw [domain = -0.5:2] plot (\\x,{pow(9,-\\x)}) (-0.5,3) node [above] {\\textcircled{2}};\n\\draw [domain = -2:0.5] plot (\\x,{pow(9,\\x)}) (0.5,3) node [above] {\\textcircled{3}};\n\\draw [domain = -2:1] plot (\\x,{pow(3,\\x)}) (1,3) node [above] {\\textcircled{4}};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$b=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.8:2] plot (\\x,{0.8*\\x+0.45});\n\\draw [domain = -1.6:2] plot (\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.3:0.8] plot (\\x,{1.8*\\x+1.45});\n\\draw [domain = -1.6:2] plot (\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1:1.3] plot (\\x,{-1.6*\\x+1.3});\n\\draw [domain = -1.6:2] plot (-\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -2:2] plot (\\x,{-0.5*\\x+0.3});\n\\draw [domain = -1.6:2] plot (\\x,{pow(0.45,0.8*\\x)});\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323243,13 +323321,15 @@ "id": "020443", "content": "利用指数函数的性质比较下列各题中两个数的大小:\\\\\n(1) $1.4^{0.3}$与$1.4^{0.4}$;\\\\\n(2) $0.3^{1.4}$与$0.3^{1.5}$;\\\\\n(3) $a^{-3.14}$与$(\\dfrac 1a)^\\pi(a>0$且$a \\neq 1)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323262,13 +323342,15 @@ "id": "020444", "content": "已知$m, n$分别满足下列条件, 比较$m, n$的大小:\\\\\n(1) $(\\dfrac 54)^{-m}<(\\dfrac 54)^{-n}$;\\\\\n(2) $(0.7)^{\\frac 1m}>(0.7)^{\\frac 1n}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323281,13 +323363,15 @@ "id": "020445", "content": "求下列不等式的解集:\\\\\n(1) $(\\dfrac 13)^{x^2-2 x+3}<(\\dfrac 13)^{2 x}$;\\\\\n(2) $4^x+3 \\cdot 2^{x+1}-16>0$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323300,13 +323384,15 @@ "id": "020446", "content": "若不等式$(2 a^2-1)^x<1$的解集为$(-\\infty, 0)$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323319,13 +323405,15 @@ "id": "020447", "content": "若函数$y=a^x$(其中$a>0$且$a \\neq 1$)在区间$[-1,2]$上的最大值为$4$, 最小值为$m$, 且函数$y=(1-4 m) \\sqrt x$在$[0,+\\infty)$上是严格增函数, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323338,13 +323426,15 @@ "id": "020448", "content": "已知$m, n$为常数, 且$a=0.9^m \\cdot 0.8^n$, $b=0.9^n \\cdot 0.8^m$, 试比较$a, b$的大小.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323357,13 +323447,15 @@ "id": "020449", "content": "已知指数函数$y=a^x$($a>0$且$a \\neq 1$)在区间$[1,2]$上的最大值比最小值大$\\dfrac{2 a}3$, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323376,13 +323468,15 @@ "id": "020450", "content": "已知$x>y>1$, 且$0a^{\\frac 1y}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323395,13 +323489,15 @@ "id": "020451", "content": "函数值域为$(0,+\\infty)$的是\\bracket{20}.\n\\fourch{$y=\\dfrac 1{x-2}$}{$y=3^{x-1}$}{$y=\\sqrt{2^x-1}$}{$y=\\sqrt{5-3x}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323414,13 +323510,15 @@ "id": "020452", "content": "函数$y=\\sqrt {(\\dfrac 1{16})^x-64}$的定义域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323433,13 +323531,15 @@ "id": "020453", "content": "已知集合$M=\\{y | y=2^x,\\ x \\in \\mathbf{R}\\}$, 集合$N=\\{y | y=x^2, \\ x \\in \\mathbf{R}\\}$, 求$M \\cap N$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323452,13 +323552,15 @@ "id": "020454", "content": "某服装店对原价分别为$175$元和$200$元的甲乙两种服装搞促销活动, 规定甲服装每天降价$5 \\%$, 直到其售完为止; 乙服装每天降价$7 \\%$, 直到其售完为止. 假设两种服装在$10$天内均没有售完, 几天后甲服装的售价将高于乙服装的售价?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323471,13 +323573,15 @@ "id": "020455", "content": "已知$0 \\leq x \\leq 2$, 求函数$y=4^{x-\\frac 12}-3 \\cdot 2^x+5$的最大值, 并求此时$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323490,13 +323594,15 @@ "id": "020456", "content": "方程$a^x+1=-x^2+2 x+2 a$($a>0$, $a \\neq 1$)的解的个数是\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323509,13 +323615,15 @@ "id": "020457", "content": "作出下列函数的大致图像:\\\\\n(1) $y=2^{|x|}$;\\\\\n(2) $y=2^{-|x|}$;\\\\\n(3) $y=|2^x-1|$;\\\\\n(4) $y=2^{|x|+1}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323528,13 +323636,15 @@ "id": "020458", "content": "设函数$f(x)=2^x-1$, $x \\in \\mathbf{R}$.\\\\\n(1) 分别作出函数$y=f(|x|)$与$y=|f(x) |$的图像;\\\\\n(2) 求实数$a$的取值范围, 使得方程$f(|x|)=a$与$|f(x)|=a$都有且仅有两个实数解.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323547,13 +323657,15 @@ "id": "020459", "content": "下列函数是对数函数的序号为\\blank{50}.(请填入全部正确序号)\\\\\n\\textcircled{1} $y=\\log_2 x$, \\textcircled{2} $y=2 \\log_2 x$,\\textcircled{3} $y=\\log_2 x^2$, \\textcircled{4} $y=\\log_2(x+2)$,\\textcircled{5} $y=\\log_2(2 x)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323566,13 +323678,15 @@ "id": "020460", "content": "已知常数$a>0$且$a \\neq 1$. 若无论$a$取何值, 函数$y=\\log_a(2 x+1)$恒经过一个定点, 则此定点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323585,13 +323699,15 @@ "id": "020461", "content": "若对数函数$y=\\log_a x$($a>0$且$a \\neq 1$)的图像经过点$(4,2)$, 则此对数函数的表达式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323604,13 +323720,15 @@ "id": "020462", "content": "写出下列函数的定义域:\\\\\n(1) $y=\\log_2 \\dfrac{2+x}{1-x}$:\\blank{50};\\\\\n(2) $y=\\log_a(4-x^2)$(常数$a>0$且$a \\neq 1$):\\blank{50};\\\\\n(3) $y=\\lg (x-2)+\\lg (10 x-3-3 x^2)$:\\blank{50};\\\\\n(4) $y=\\dfrac{\\sqrt {2 x-1}}{\\lg x}$:\\blank{50};\\\\\n(5) $y=\\log_{x-1}(3-x)$:\\blank{50};\\\\\n(6) $y=\\sqrt {\\lg x}+\\lg (5-2 x)$:\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323623,13 +323741,15 @@ "id": "020463", "content": "在同一坐标系中作出$y=\\lg x$及$y=\\log_{0.1} x$的大致图像.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323642,13 +323762,15 @@ "id": "020464", "content": "已知集合$A, B$分别是函数$y=x^{-\\frac 12}$与$y=\\lg (1-2 x)$的定义域, 则$A \\cap B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323661,13 +323783,15 @@ "id": "020465", "content": "$a, b, c$是图中三个对数函数的底数, 它们之间的大小关系是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw [->] (0,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(2)}) (3,{ln(3)/ln(2)}) node [right] {$y=\\log_c x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(0.4)}) (3,{ln(3)/ln(0.4)}) node [right] {$y=\\log_b x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(0.6)}) (3,{ln(3)/ln(0.6)}) node [below left] {$y=\\log_a x$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$c>a>b$}{$c>b>a$}{$a>b>c$}{$b>a>c$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323680,13 +323804,15 @@ "id": "020466", "content": "判断函数$y=\\ln|x|$的图像是否关于某条直线对称? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323699,13 +323825,15 @@ "id": "020467", "content": "下列函数是对数函数的序号为\\blank{50}. (请填入全部正确序号)\\\\\n\\textcircled{1} $y=\\log_2 x$, \\textcircled{2} $y=2 \\log_2 x$, \\textcircled{3} $y=\\log_2 x^2$, \\textcircled{4} $y=\\log_2(x+2)$, \\textcircled{5} $y=\\log_2(2 x)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323718,13 +323846,15 @@ "id": "020468", "content": "已知常数$a>0$且$a \\neq 1$. 若无论$a$取何值, 函数$y=\\log_a(2 x+1)$恒经过一个定点, 则此定点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323737,13 +323867,15 @@ "id": "020469", "content": "若函数$y=\\log_a(x+1)$是严格减函数, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323756,13 +323888,15 @@ "id": "020470", "content": "写出下列函数的定义域:\\\\\n(1) $y=\\lg (x-2)+\\lg (10 x-3-3 x^2)$;\\\\\n(2) $y=\\dfrac{\\sqrt {2 x-1}}{\\lg x}$;\\\\\n(3) $y=\\log_{x-1}(3-x)$;\\\\\n(4) $y=\\sqrt {\\lg x}+\\lg (5-2 x)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323775,13 +323909,15 @@ "id": "020471", "content": "根据下列不等式, 确定实数$a$的取值范围:\\\\ \n(1) $\\log_a 0.2<\\log_a 0.1$;\\\\\n(2) $\\log_a \\pi>\\log_a \\mathrm{e}$;\\\\\n(3) $\\log_a 3<0$;\\\\\n(4) $\\log_a \\dfrac 45<1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323794,13 +323930,15 @@ "id": "020472", "content": "已知$10$且$a \\neq 1$. 若无论$a$取何值, 函数$y=4+\\log_a(x^2-x-1)$的图像恒经过定点$P$, 求点$P$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323832,13 +323972,15 @@ "id": "020474", "content": "已知集合$P$是函数$y=\\dfrac 1{\\sqrt {1-x^2}}$的定义域, 集合$Q$是函数$y=\\log_{\\frac 12}(2+x-6 x^2)$的定义域, 求$P \\cap Q$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323851,13 +323993,15 @@ "id": "020475", "content": "已知常数$a \\in \\mathbf{R}$, 集合$A=\\{x | 2^{x^2-x-6}>1\\}$, 集合\n$B=\\{x | \\log_4(x+1)=latex,scale = 0.7]\n\\draw [->] (0,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(1.8)}) (3,{ln(3)/ln(1.8)}) node [above left] {$y=\\log_a x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(2)}) (3,{ln(3)/ln(2)}) node [right] {$y=\\log_b x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(0.4)}) (3,{ln(3)/ln(0.4)}) node [right] {$y=\\log_c x$};\n\\draw [domain = 0.3:5] plot (\\x,{ln(\\x)/ln(0.6)}) (3,{ln(3)/ln(0.6)}) node [below left] {$y=\\log_d x$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$a>b>c>d$}{$a>b>d>c$}{$b>a>c>d$}{$b>a>d>c$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323927,13 +324077,15 @@ "id": "020479", "content": "函数$y=\\log_a x$在区间$[a, 2 a]$上的最大值与最小值的差为$\\dfrac 12$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323946,13 +324098,15 @@ "id": "020480", "content": "设$a>0$且$a \\neq 1$, 比较$\\log_a 2 a$与$\\log_a 3 a$的大小.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323965,13 +324119,15 @@ "id": "020481", "content": "已知常数$k \\in \\mathbf{R}$, 函数$y=\\log_4(4^x+1)-(k-1) x(x \\in \\mathbf{R})$关于$y$轴对称.\\\\\n(1) 求常数$k$的值;\\\\\n(2) 当$x$取何值时函数值最小? 并求出该函数的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -323984,13 +324140,15 @@ "id": "020482", "content": "函数$y=\\log_{0.5}(3-2 x-x^2)$的递增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324003,13 +324161,15 @@ "id": "020483", "content": "函数$y=\\log_{0.5}^2 x+4 \\log_{0.5} x-3$的递增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324022,13 +324182,15 @@ "id": "020484", "content": "函数$y=\\log_{\\frac 12}(x^2-6 x+7)$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324041,13 +324203,15 @@ "id": "020485", "content": "不等式$\\log_{\\frac 12}(4+3 x-x^2)>\\log_{\\frac 12}(4 x-2)$的解是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324060,13 +324224,15 @@ "id": "020486", "content": "函数$f(x)=\\log_a x$在区间$[a, 2 a]$上的最大值与最小值之差为$\\dfrac 12$, 则$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324079,13 +324245,15 @@ "id": "020487", "content": "设$f(x)=|\\lg x|$, 当$0f(p)>f(n)$, 则\\bracket{20}.\n\\fourch{$(m-1)(p-1)>0$}{$m p>1$}{$m p=1$}{$m p<1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324098,13 +324266,15 @@ "id": "020488", "content": "若改为函数$y=\\log_a(2-a x)$在$[1,3]$上是增函数, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324117,13 +324287,15 @@ "id": "020489", "content": "若函数$f(x)=\\lg (k x^2-\\mathbf{4 x}+k+3)$的定义域是$R$, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324136,13 +324308,15 @@ "id": "020490", "content": "若函数$f(x)=\\lg (k x^2-4 x+k+3)$的值域是$\\mathbf{R}$, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324155,13 +324329,15 @@ "id": "020491", "content": "已知$x$满足不等式$\\log_2^2 x-5 \\log_2 x+6 \\leq 0$, 求函数$f(x)=\\log_2 \\dfrac x2 \\log_2 \\dfrac x4$的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324174,13 +324350,15 @@ "id": "020492", "content": "函数$y=\\log_{\\frac 12}(x^2-6 x+17)$的值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324193,13 +324371,15 @@ "id": "020493", "content": "若$0>\\log_m 2>\\log_n 2$, 则实数$m, n$的大小关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324212,13 +324392,15 @@ "id": "020494", "content": "函数$y=\\sqrt {\\log_{0.2} \\dfrac{2 x-1}{3-x}}$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324231,13 +324413,15 @@ "id": "020495", "content": "求下列函数的定义域及值域:\\\\\n(1) $y=\\log_{\\frac 12} \\sqrt {-x^2+2 x+1}$;\\\\\n(2) $y=\\sqrt {\\lg (-x^2+6 x+7)}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324250,13 +324434,15 @@ "id": "020496", "content": "函数$f(x)=\\log_9(x+8-\\dfrac ax)$在$[1,+\\infty)$上是增函数, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324269,13 +324455,15 @@ "id": "020497", "content": "设$f(x)=x^2-x+k$, 若$f(\\log_2 a)=k$以及$\\log_2 f(a)=2$, 其中$a, k$为常数, 且$a \\neq 1$, 求$f(\\log_2 x)$的最小值以及相应的$x$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第四章试题", + "origin": "2025届高一校本作业必修第四章", "edit": [ "20221230\t王伟叶" ], @@ -324288,13 +324476,15 @@ "id": "020498", "content": "下列各组函数$f(x)$与$g(x)$表示同一个函数的是\\bracket{20}.\n\\twoch{$f(x)=\\dfrac{x^2-1}{x+1}$, $g(x)=x-1$}{$f(x)=|x|$, $g(x)=\\begin{cases}x, & x \\geq 0,\\\\ -x, & x<0 \\end{cases}$}{$f(x)=x^0$, $g(x)=1$}{$f(x)=(\\sqrt x)^2$, $g(x)=\\sqrt {x^2}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324307,13 +324497,15 @@ "id": "020499", "content": "已知镭经过$100$年后剩下原来质量的$95.76\\%$, 若质量为$1$克的镭经过$x$年后的剩余质量为$y$克, 则$y$与$x$之间的解析式是\\bracket{20}.\n\\twoch{$y=(\\dfrac{0.9576}{100})^x$}{$y=(0.9576)^{100 x}$}{$y=(0.9576)^{\\frac x{100}}$}{$y=1-(1-0.9576)^{\\frac x{100}}$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324326,13 +324518,15 @@ "id": "020500", "content": "若$f(x)=x^2+p x+q$, 且$f(1)=0, f(2)=0$, 求$f(-1)$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324345,13 +324539,15 @@ "id": "020501", "content": "下列各图像中, 哪些是函数的图像, 哪些不是函数的图像? 为什么?\\\\\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.9]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = {-pi/2}:{pi/2}] plot ({\\x},{-sin(2*\\x/pi*180)});\n\\draw (0,-2) node [below] {(1)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.9]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw [domain = {-pi/2}:{pi/2}] plot ({sin(2*\\x/pi*180)},{\\x});\n\\draw (0,-2) node [below] {(2)}; \n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0,1,2,3} \n{ \n\\draw (\\i,{\\i+1}) -- ({\\i+1},{\\i+1});\n\\filldraw [fill = white, draw = black] (\\i,{\\i+1}) circle (0.05);\n\\filldraw ({\\i+1},{\\i+1}) circle (0.05);\n};\n\\filldraw [fill = white, draw = black] (0,0) circle (0.05);\n\\foreach \\i in {1,2,3,4}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$};\n};\n\\draw (2,-1) node [below] {(3)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {0,1,2,3} \n{ \n\\draw ({\\i+1},\\i) -- ({\\i+1},{\\i+1});\n\\filldraw [fill = white, draw = black] ({\\i+1},{\\i+1}) circle (0.05);\n\\filldraw ({\\i+1},\\i) circle (0.05);\n};\n\\filldraw [fill = white, draw = black] (0,0) circle (0.05);\n\\foreach \\i in {1,2,3,4}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$};\n};\n\\draw (2,-1) node [below] {(4)};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324364,13 +324560,15 @@ "id": "020502", "content": "求函数求函数$y=\\dfrac 1{\\sqrt {x^2+2 x+3}}$的值域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324383,13 +324581,15 @@ "id": "020503", "content": "观察下列各函数, 并写出他们的值域: \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-1,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2,3}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$}; \n};\n\\draw (0,0) -- (1,1) -- (2,1) (2,2) -- (3,2);\n\\draw [dashed] (0,1) -- (1,1) -- (1,0) (0,2) -- (2,2) (2,0) -- (2,2) (3,0) -- (3,2);\n\\filldraw [fill = white, draw = black] (2,1) circle (0.05) (2,2) circle (0.05) (3,2) circle (0.05);\n\\draw (1.5,-1) node [below] {(1)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (2.5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$}; \n};\n\\foreach \\i in {-1,-2}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n};\n\\draw (-2,2) -- (-1,2) -- (0,0) -- (1,2) -- (2,2);\n\\draw [dashed] (-2,0) -- (-2,2) (2,0) -- (2,2) (-1,2) -- (1,2);\n\\filldraw [fill = white, draw = black] (-2,2) circle (0.05) (2,2) circle (0.05);\n\\draw (0,-1) node [below] {(2)};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-2.5,0) -- (5.5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {-2,-1,1,2,3,4,5}\n{\n\\draw (\\i,0.1) -- (\\i,0) node [below] {$\\i$};\n};\n\\foreach \\i in {-1,1,2,3}\n{\n\\draw (0.1,\\i) -- (0,\\i) node [left] {$\\i$}; \n};\n\\draw [domain = -2:5] plot (\\x,{2/49*(3+2*sqrt(2))*\\x*\\x+4/49*(-1-3*sqrt(2))*\\x+(17-40*sqrt(2))/49});\n\\draw [dashed] (0,-1) -- ({7*sqrt(2)-9},-1) (5,0) -- (5,3) -- (0,3) (-2,0) -- (-2,1) -- (0,1);\n\\filldraw [fill = white, draw = black] (-2,1) circle (0.05) (5,3) circle (0.05);\n\\draw (1.5,-1) node [below] {(3)};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324402,13 +324602,15 @@ "id": "020504", "content": "若函数$f(x)=\\lg (\\dfrac{1+x}{1-x})$($-18 \\\\f(x+3), & x \\leq 8,\\end{cases}$ 则$f(2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324535,13 +324749,15 @@ "id": "020511", "content": "如果函数$f(x)=\\dfrac{x-1}{x-2}$, 那么$f(x+1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324554,13 +324770,15 @@ "id": "020512", "content": "如果函数$f(x)=2 x^2-1$, 那么$f(f(x))=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324573,13 +324791,15 @@ "id": "020513", "content": "已知函数$f(x)$的定义域为$[1,4]$, 则$f(x+1)+f(x-1)$的定义域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324592,13 +324812,15 @@ "id": "020514", "content": "已知函数$y=\\begin{cases}x^2-5, & x \\leq 0, \\\\-x^2, & x>0.\\end{cases}$\\\\\n(1) 求$f(2)$、$f[f(-2)]$的值;\\\\ \n(2) 若$f(a)=-1$, 求$a$的值;\\\\\n(3) 若$f(a)>-1$, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324611,13 +324833,15 @@ "id": "020515", "content": "函数$f(x)=\\begin{cases}-x, & -10$, 均成立$f(x^6)=\\log_2 x$, 则$f(8)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324668,13 +324896,15 @@ "id": "020518", "content": "已知函数$y=f(x)$满足: 对一切实数$x$, 均成立$f(x+1)=x^2-3 x+2$, 求$f(x)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324687,13 +324917,15 @@ "id": "020519", "content": "已知定义域为$\\{x|x\\ne -2, \\ x \\in \\mathbf{R}\\}$的函数$y=f(x)$满足: 对一切$x\\in \\{x|x\\ne -3, \\ x\\in \\mathbf{R}\\}$, 均成立$f(\\dfrac{1-2 x}{3+x})=3 x-1$, 求$f(x)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324706,13 +324938,15 @@ "id": "020520", "content": "正方形$ABCD$的边长为$2$, 动点$P$从点$A$出发, 沿着正方形的边界经过点$B, C, D$回到$A$点, 设$P$经过的路程为$x$, 点$P$到正方形对角线$AC$的距离$PQ$为$y$. 将$y$表示成$x$的函数, 并画出函数图像.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\draw (2,2) node [above right] {$C$} coordinate (C);\n\\draw (0,2) node [above left] {$D$} coordinate (D);\n\\draw (1.4,0) node [below] {$P$} coordinate (P);\n\\draw ($(A)!(P)!(C)$) node [above left] {$Q$} coordinate (Q);\n\\draw (A) rectangle (C);\n\\draw (P) -- (Q) (A) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324725,13 +324959,15 @@ "id": "020521", "content": "若函数$f(x)$是奇函数, 则$f(1+\\sqrt 2)+f(\\dfrac 1{1-\\sqrt 2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324744,13 +324980,15 @@ "id": "020522", "content": "已知函数$f(x)=a x^5+b x^{\\frac 13}+2$. 若$f(2)=3$, 则$f(-2)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324763,13 +325001,15 @@ "id": "020523", "content": "函数$y=\\sqrt {x^2-4}+\\sqrt {4-x^2}$的奇偶性是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324782,13 +325022,15 @@ "id": "020524", "content": "若函数$y=f(x)$的定义域为$\\mathbf{R}$, 则$y=f(x)$为奇函数的充要条件为\\bracket{20}.\n\\onech{$f(0)=0$}{对任意$x \\in \\mathbf{R}, f(x)=0$}{存在某个$x_0 \\in \\mathbf{R}$, 使得$f(x_0)+f(-x_0)=0$}{对任意的$x \\in \\mathbf{R}$, $f(x)+f(-x)=0$都成立}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324801,13 +325043,15 @@ "id": "020525", "content": "``函数$f(x)$的定义域关于原点对称''是``函数$f(x)$为奇函数''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324820,13 +325064,15 @@ "id": "020526", "content": "下列命题中, 真命题是\\bracket{20}.\n\\onech{偶函数的图像一定与$y$轴相交}{奇函数的图像一定通过原点}{偶函数的图像关于$y$轴对称}{即是奇函数又是偶函数的函数只能是$f(x)=0$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324839,13 +325085,15 @@ "id": "020527", "content": "判断下列函数$y=f(x)$的奇偶性, 并给出证明:\\\\\n(1) $f(x)=2 x+\\sqrt[3]x$;\\\\\n(2) $f(x)=2 x-x^2$;\\\\\n(3) $f(x)=x^2-x^4$;\\\\\n(4) $f(x)=\\dfrac{1-x}{1+x}$;\\\\\n(5) $f(x)=\\lg \\dfrac{1-x}{1+x}$;\\\\\n(6) $f(x)=\\begin{cases}-x^2+x, & x \\geq 0, \\\\x^2+x, & x<0.\\end{cases}$", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324858,13 +325106,15 @@ "id": "020528", "content": "已知$f(x)$为偶函数, 当$x \\geq 0$时, $f(x)=(x-1)^3+1$, 求$f(x)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324877,13 +325127,15 @@ "id": "020529", "content": "若函数$y=f(x)$, $x \\in D$为非奇非偶函数, 则有\\bracket{20}.\n\\onech{对于任意$x_0 \\in D$, 都有$f(-x_0) \\neq f(x_0)$且$f(-x_0) \\neq-f(x_0)$}{存在$x_0 \\in D$, 使$f(-x_0) \\neq f(x_0)$且$f(-x_0) \\neq-f(x_0)$}{存在$x_1$、$x_2 \\in D$, 使$f(-x_1) \\neq f(x_1)$且$f(-x_2) \\neq-f(x_2)$}{对于任意$x_0 \\in D$, 都有$f(-x_0) \\neq f(x_0)$或$f(-x_0) \\neq-f(x_0)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324896,13 +325148,15 @@ "id": "020530", "content": "判断下列函数的奇偶性, 并给出证明:\\\\\n(1) $y=x(\\dfrac 1{2^x-1}+\\dfrac 12)$;\\\\\n(2) $y=\\dfrac{\\sqrt {1-x^2}}{|x+2|-2}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324915,13 +325169,15 @@ "id": "020531", "content": "已知函数$f(x)=x^2+a x+1, x \\in[b, 2]$是偶函数, 求$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324934,13 +325190,15 @@ "id": "020532", "content": "是否存在一个实数$a$, 使得函数$y=\\log_2 \\dfrac{a-x}{2+x}$是奇函数? 若存在, 求$a$的值; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324953,13 +325211,15 @@ "id": "020533", "content": "若$f(x)$为奇函数, 则$f(x)+f(-x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324972,13 +325232,15 @@ "id": "020534", "content": "若$f(x)$为偶函数, 则$f(\\sqrt 2+1)-f(\\dfrac 1{1-\\sqrt 2})=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -324991,13 +325253,15 @@ "id": "020535", "content": "已知$f(x)=a x^2+b x+c$($a \\neq 0$)是偶函数, 则$g(x)=a x^3+b x^2+c x$的奇偶性为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325010,13 +325274,15 @@ "id": "020536", "content": "已知$f(x)=a x^2+b x+3 a+b$是偶函数, 且其定义域为$[a-1,2 a]$, 则$a=$\\blank{50}, $b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325029,13 +325295,15 @@ "id": "020537", "content": "设$f(x)=a x^5+b x^3+c x+1$, 若$f(-\\pi)=3$, 则$f(\\pi)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325048,13 +325316,15 @@ "id": "020538", "content": "若$f(x)=(k-1) x^2+2 k x+3$是偶函数, 则$f(-1)$、$f(-\\sqrt 2)$、$f(\\sqrt 3)$由小到大的顺序是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325067,13 +325337,15 @@ "id": "020539", "content": "判断下列函数的奇偶性:\\\\\n(1) $f(x)=|\\dfrac 12 x-3|+|\\dfrac 12 x+3|$;\\\\\n(2) $f(x)=x^2+1$, $x \\in(a, 1)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325086,13 +325358,15 @@ "id": "020540", "content": "设$f(x)$是定义在$\\mathbf{R}$上的偶函数, 当$x \\leq 0$时, $f(x)=2 x(x+1)$, 求$f(x)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325105,13 +325379,15 @@ "id": "020541", "content": "已知函数$f(x)=x^2-2 a|x-1|$, $x \\in \\mathbf{R}$, $a$为实常数.\\\\\n(1) 求证: 函数$f(x)$不是奇函数;\\\\\n(2) 若函数$f(x)$是偶函数, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325124,13 +325400,15 @@ "id": "020542", "content": "若函数$f(x)=\\dfrac 1{2^x+1}+m$为奇函数, 则$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325143,13 +325421,15 @@ "id": "020543", "content": "设$f(x)$对任意$x$、$y$满足$f(x+y)=f(x)+f(y)$, 且$f(2)=4$, 则$f(-1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325162,13 +325442,15 @@ "id": "020544", "content": "设$f(x)$是定义在$\\mathbf{R}$上的奇函数, $g(x)$是定义在$\\mathbf{R}$上的偶函数.\\\\\n(1) 判断$F(x)=[f(x)]^2-3 g(x)$的奇偶性;\\\\\n(2) 若$f(x)+g(x)=6 x^2-2 x+3$, 求$f(x), g(x)$的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325181,13 +325463,15 @@ "id": "020545", "content": "已知定义在$\\mathbf{R}$上的函数, 满足对于任意$x, y \\in \\mathbf{R}$, 都成立$f(x+y)=f(x)+f(y)$, 且$f(x)$不恒等于零.\\\\\n(1) 求$f(0)$;\\\\\n(2) 判断$f(x)$的奇偶性;\\\\\n(3) 若$f(-3)=a$, 用$a$表示$f(12)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325200,13 +325484,15 @@ "id": "020546", "content": "函数$y=-\\dfrac 1x$的单调增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325219,13 +325505,15 @@ "id": "020547", "content": "若函数$f(x)=k x^2+2 x+4$在区间$[1,+\\infty)$上是严格减函数, 在区间$(-\\infty, 1]$上是严格增函数, 则$f(3)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325238,13 +325526,15 @@ "id": "020548", "content": "若偶函数$y=f(x)$在$(-\\infty, 0)$上是严格增函数, 则$f(-2)$、$f(\\pi)$、$f(\\dfrac{\\sqrt 3}2)$按从小到大排列为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325257,13 +325547,15 @@ "id": "020549", "content": "若函数$y=2 x^2+a x+3$在区间$(-\\infty, 4]$上是严格减函数, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325276,13 +325568,15 @@ "id": "020550", "content": "已知函数$y=k x^2-4 x-8$在区间$[4,16]$上是严格减函数, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325295,13 +325589,15 @@ "id": "020551", "content": "已知函数$f(x)$同时满足下列条件: \\textcircled{1} 奇函数, \\textcircled{2} 当$x \\in[-2,2]$时, $f(x)$是严格增函数. $f(x)$可以是\\blank{50}.(写出一个满足条件的函数表达式)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325314,13 +325610,15 @@ "id": "020552", "content": "写出下列函数的单调区间:\\\\\n(1) $y=|x-1|$:\\blank{100};\\\\\n(2) $f(x)=a x+b(a \\neq 0)$:\\blank{100};\\\\\n(3) $y=x^2-4 x, x \\in[-3,3]$:\\blank{100};\\\\\n(4) $f(x)=a x^2+2 a x+a^2$, $x \\in(-\\infty,-1]$($a \\neq 0$):\\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325333,13 +325631,15 @@ "id": "020553", "content": "已知$f(x)$是定义在$(-3,3)$上的奇函数, 且该函数在$[0,3)$上的解析式为$y=(x-1)^2-1$. 求$f(x)$在$(-3,0)$上的解析式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325352,13 +325652,15 @@ "id": "020554", "content": "设$f(x)=|\\lg x|$, 三个实数$m,n,p$满足$0f(p)>f(n)$, 则\\bracket{20}.\n\\fourch{$(m-1)(p-1)>0$}{$m p>1$}{$m p=1$}{$m p<1$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325371,13 +325673,15 @@ "id": "020555", "content": "已知定义在$\\mathbf{R}$上的函数$f(x)=2^{|x-m|}-1$(其中$m$为实数)的图像关于$y$轴对称, 记$a=\\log_{0.5} 3$, $b=f(\\log_25), c=f(2 m)$则$a, b, c$的大小关系为\\bracket{20}.\n\\fourch{$af(a^2-a+1)$}{$f(-\\dfrac 34) \\geq f(a^2-a+1)$}{$f(-\\dfrac 34)0$, 则有$f(a)+f(b)>f(-a)+f(-b)$;\\\\\n其中正确命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325694,13 +326030,15 @@ "id": "020572", "content": "已知函数$f(x)=x^2+\\dfrac ax$($x \\neq 0$, 常数$a \\in \\mathbf{R}$).\\\\\n(1) 讨论函数$f(x)$的奇偶性, 并说明理由;\\\\ \n(2) 若函数$f(x)$在$x \\in[2,+\\infty)$上为严格增函数, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325713,13 +326051,15 @@ "id": "020573", "content": "已知函数$f(x)$是定义在$(0,+\\infty)$上的严格增函数, 且满足$f(x y)=f(x)+f(y)$, $f(2)=1$, 解不等式:$f(x)-f(x-2)>3$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325732,13 +326072,15 @@ "id": "020574", "content": "求下列函数的最大值或最小值, 并求出取最值时相应的自变量$x$的值.\\\\\n(1) $f(x)=x^2-4 x-2$;\\\\\n(2) $f(x)=6 x-3 x^2$;\\\\\n(3) $f(x)=-x^2-4 x-3$, $x \\in[-3,1]$;\\\\\n(4) $f(x)=x^2-2 x-3$, $x \\in[-2,0]$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325751,13 +326093,15 @@ "id": "020575", "content": "写出下列函数的值域:\\\\\n(1) $y=1-\\dfrac 1{x+2}$:\\blank{50};\\\\\n(2) $y=\\dfrac{x+3}{x-3}$, $x \\in[0,1)$:\\blank{50};\\\\\n(3) $y=\\dfrac{5 x+3}{x-3}$, $x \\in(-\\infty, 2) \\cup[4,+\\infty)$:\\blank{50};\\\\\n(4) $y=4+\\sqrt {2 x+1}$:\\blank{50};\\\\\n(5) $y=\\sqrt {x-\\dfrac 12 x^2}$:\\blank{50};\\\\\n(6) $y=\\sqrt {-x^2+x+2}$:\\blank{50};\\\\\n(7) $y=\\dfrac{2 x^2+2 x+3}{x^2+x+1}$:\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325770,13 +326114,15 @@ "id": "020576", "content": "已知$p, q$分别是函数$f(x)=-2 x+3$在$[-2,2]$上的最大值和最小值, 求函数$g(x)=2 x^2-p x+q$在$[-2,2]$上的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325789,13 +326135,15 @@ "id": "020577", "content": "若函数$f(x)=a x^2-2 a x+2+b$在$[2,3]$上有最大值$5$、最小值$2$, 求$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325808,13 +326156,15 @@ "id": "020578", "content": "函数$y=\\sqrt {\\lg (-3 x^2+6 x+7)}$的定义域是\\blank{50}; 值域是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325827,13 +326177,15 @@ "id": "020579", "content": "函数$y=\\log_{\\frac 15}(x^2-6 x+10)$在区间$[1,2]$上的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325846,13 +326198,15 @@ "id": "020580", "content": "求下列函数的定义域和值域:\\\\\n(1) $y=4^x-2^{x+1}+1$($x<0$);\\\\\n(2) $y=\\sqrt {1-(\\dfrac 12)^x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325865,13 +326219,15 @@ "id": "020581", "content": "已知$\\alpha$、$\\beta$是方程$4 x^2-4 m x+m+2=0$的两个实数根, 当$m$为何值时, $\\alpha^2+\\beta^2$有最小值? 并求出这个最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325884,13 +326240,15 @@ "id": "020582", "content": "求函数$y=x^2-4 x+1$在$x \\in[t, 4]$上的最小值和最大值, 其中$t<4$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325903,13 +326261,15 @@ "id": "020583", "content": "已知集合$A=\\{x | 1 \\leq x \\leq 4\\}$, $f(x)=x^2+p x+q$和$g(x)=x+\\dfrac 4x$是定义$A$上的函数, 且在$x_0$处同时取到最小值, 并满足$f(x_0)=g(x_0)$, 求$f(x)$在$A$上的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325922,13 +326282,15 @@ "id": "020584", "content": "函数$f(x)=\\dfrac 2x$的单调减区间为\\blank{50}, 奇偶性为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325941,13 +326303,15 @@ "id": "020585", "content": "函数$f(x)=x^2+x$的单调增区间为\\blank{50}, 奇偶性为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325960,13 +326324,15 @@ "id": "020586", "content": "函数$f(x)=a x+b(a>0)$的单调区间为\\blank{50}, 当$b=$\\blank{50}时函数$f(x)$为奇函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325979,13 +326345,15 @@ "id": "020587", "content": "若函数$f(x)=x^2+k x+1$在区间$[1,+\\infty)$上是严格增函数, 在区间$(-\\infty, 1]$上是严格减函数, 则实数$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -325998,13 +326366,15 @@ "id": "020588", "content": "若函数$f(x)=x^2+k x+1$是偶函数, 则实数实数$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326017,13 +326387,15 @@ "id": "020589", "content": "若函数$y=2 x^2+a x+3$在区间$(-\\infty, 4]$上是严格减函数, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326036,13 +326408,15 @@ "id": "020590", "content": "写出下列函数的单调区间, 奇偶性, 并作出大致图像.\\\\\n(1) $y=x^2-4 x$, $x \\in[-3,3]$;\\\\\n(2) $f(x)=2 x$, $x \\in[-1,1)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326055,13 +326429,15 @@ "id": "020591", "content": "证明: 函数$f(x)=\\dfrac 1x$不是其定义域上的单调函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326074,13 +326450,15 @@ "id": "020592", "content": "已知$a$、$b$为常数, 函数$f(x)=x^2+a x+1$, $x \\in[b, 2]$是偶函数, 求$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326093,13 +326471,15 @@ "id": "020593", "content": "已知函数$y=a x+b$($a, b \\in \\mathbf{R}$且$a>0)$.\\\\ \n(1) 证明函数$y=a x+b$($a, b \\in \\mathbf{R}$且$a>0$)严格递增;\\\\\n(2) 判断函数的奇偶性, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326112,13 +326492,15 @@ "id": "020594", "content": "已知函数$y=\\dfrac 1x$在区间$[-1, a]$上严格递减, 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326131,13 +326513,15 @@ "id": "020595", "content": "已知函数$y=k x^2-4 x-8$在区间$[4,16]$上是严格减函数, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326150,13 +326534,15 @@ "id": "020596", "content": "已知$f(x)$是定义在$(-3,3)$上的奇函数, 且该函数在$[0,3)$上的解析式为$y=(x-1)^2-1$.\\\\\n(1) 求$f(x)$在$(-3,0)$上的解析式;\\\\\n(2) 在平面直角坐标系中作出函数$f(x)$的图像, 并写出单调区间.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326169,13 +326555,15 @@ "id": "020597", "content": "求函数$y=x^2+a x+1$($a \\in \\mathbf{R}$)的最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326188,13 +326576,15 @@ "id": "020598", "content": "求函数$f(x)=x^2-4 x+1$, $x \\in[t, 4]$的最小值和最大值, 其中$t<4$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326207,13 +326597,15 @@ "id": "020599", "content": "已知$a \\in \\mathbf{R}$, 求函数$f(x)=x^2+a x+3$, $x \\in[2,3]$的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326226,13 +326618,15 @@ "id": "020600", "content": "若函数$f(x)=a x^2-2 a x+2+b$在$[2,3]$上有最大值$5$、最小值$2$, 求$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326245,13 +326639,15 @@ "id": "020601", "content": "已知$a \\in \\mathbf{R}$, 求函数$f(x)=-x^2+2 x+1$, $x \\in[a, a+2]$的最大值和最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326264,13 +326660,15 @@ "id": "020602", "content": "已知函数$f(x)=4^{-x}-a \\cdot 2^{1-x}-3$, $x \\in[-2,+\\infty)$的最小值是$-4$, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326283,13 +326681,15 @@ "id": "020603", "content": "已知常数$a \\in \\mathbf{R}$, 函数$f(x)=x^2-2 a x+3$.\\\\\n(1) 若函数的图像都在$x$轴的上方, 求$a$的取值范围;\\\\\n(2) 当$x \\in[0,+\\infty)$时, $f(x) \\geq 0$恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326302,13 +326702,15 @@ "id": "020604", "content": "已知$\\alpha$、$\\beta$是方程$4 x^2-4 m x+m+2=0(m \\in \\mathbf{R})$的两个实数根(包含重根), 求$\\alpha^2+\\beta^2$的最小值, 并指出此时$m$取何值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326321,13 +326723,15 @@ "id": "020605", "content": "将函数$y=\\dfrac 1{2 x+3}$的图像向右平移$2$个单位后, 所得图像的函数解析式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326340,13 +326744,15 @@ "id": "020606", "content": "函数$y=\\dfrac{x+4}{x+3}$的图像可由函数$y=\\dfrac 1x$的图像先向左平移\\blank{20}个单位, 再向上平移\\blank{20}个单位.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326359,13 +326765,15 @@ "id": "020607", "content": "将函数$y=x^2$($-1 \\leq x \\leq 1$)的图像先向左平移$2$个单位, 再向下平移$3$个单位后, 所得图像的函数解析为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326378,13 +326786,15 @@ "id": "020608", "content": "函数$y=f(x-3)+1$的图像可以由函数$y=f(x)$图像先向\\blank{20}平移$3$个单位, 再向\\blank{20}平移$1$个单位得到.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326397,13 +326807,15 @@ "id": "020609", "content": "函数$f(x)=x^2+1$的图像向上平移$1$个单位, 再关于$x$轴对称, 可以得到函数\\blank{50}的图像.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326416,13 +326828,15 @@ "id": "020610", "content": "函数$y=f(-x+1)$的图像可以由函数$y=f(x)$的图像先向\\blank{20}平移$1$个单位, 再关于\\blank{20}对称得到.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326435,13 +326849,15 @@ "id": "020611", "content": "函数$y=-x^2+|x|+1$的图像可以由二次函数$y=x^2-x$经过怎样的变换得到?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326454,13 +326870,15 @@ "id": "020612", "content": "作出下列函数的图像:\\\\\n(1) $y=\\dfrac{2 x-3}{x-3}$;\\\\\n(2) $y=\\dfrac 1{|x+2|-1}$;\\\\\n(3) $y=1-\\log_2 x$;\\\\\n(4) $y=|2^x-1|$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326473,13 +326891,15 @@ "id": "020613", "content": "试作出函数$y=x+\\dfrac 1{x-1}$的图像, 并指出函数图像关于哪一点对称?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326492,13 +326912,15 @@ "id": "020614", "content": "已知函数$f(x)=|x^2-2 x|$.\\\\\n(1) 作出函数$f(x)$的图像;\\\\\n(2) 写出$f(x)$的性质(包括定义域、值域、单调区间、最值).", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326511,13 +326933,15 @@ "id": "020615", "content": "设函数图像$C$对应的函数为$y=x^3-x$, 将$C$沿$x$轴、$y$轴正方向分别平移$t$、$s$($s,t>0$)个单位长度后得到函数图象$C_1$.\\\\ \n(1) 写出函数图象$C_1$所对应的函数表达式;\\\\\n(2) 证明: 函数图象$C$与$C_1$关于点$A(\\dfrac t2, \\dfrac s2)$对称.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326530,13 +326954,15 @@ "id": "020616", "content": "函数$y=2 x+\\dfrac 1x$的奇偶性为\\blank{50}, 递增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326549,13 +326975,15 @@ "id": "020617", "content": "若函数$f(x)=x+\\dfrac 4x$($1 \\leq x \\leq 5$), 则函数$y=f(x)$的递减区间是\\blank{50}, 最小值是\\blank{50}, 最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326568,13 +326996,15 @@ "id": "020618", "content": "设常数$b \\in \\mathbf{R}$. 若函数$y=x+\\dfrac bx$($x>0$)在$(0,4]$上是严格减函数, 在$[4,+\\infty)$上是严格增函数, 则$b=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326587,13 +327017,15 @@ "id": "020619", "content": "函数$y=x+\\dfrac 5{x-2}$($x \\leq 0$)的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326606,13 +327038,15 @@ "id": "020620", "content": "函数$y=2 x^2+\\dfrac{16}{x^2+1}$, $x \\in[1,2]$取最大值时$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326625,13 +327059,15 @@ "id": "020621", "content": "已知$a>0$, 函数$f(x)=x-\\dfrac ax$, 则函数$y=f(x)$的递增区间是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326644,13 +327080,15 @@ "id": "020622", "content": "设常数$p, q \\in \\mathbf{R}$, 若在区间$[\\dfrac 12, 2]$上, 函数$f(x)=x^2+p x+q$与$g(x)=x+\\dfrac 1x$在同一点取得相同的最小值, 求$f(x)$在$[\\dfrac 12, 2]$上的最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326663,13 +327101,15 @@ "id": "020623", "content": "若$c \\in \\mathbf{R}$, 函数$f(x)=x+\\dfrac cx$($1 \\leq x \\leq 2$), 请写出函数$f(x)$单调区间, 并求函数的最值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326682,13 +327122,15 @@ "id": "020624", "content": "设常数$a, m \\in \\mathbf{R}$. 已知函数$f(x)=\\dfrac{x^2+2 x+a}x$($x \\geq m$且$x \\neq 0$).\\\\\n(1) 设$a=\\dfrac 12$, 求函数$y=f(x)$的值域;\\\\\n(2) 设$m=1$, 求函数$y=f(x)$的值域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326701,13 +327143,15 @@ "id": "020625", "content": "已知$\\dfrac 13 \\leq a \\leq 1$, 若函数$f(x)=a x^2-2 x+1$在$x \\in[1,3]$上的最小值为$N(a)$, 最大值为$M(a)$, 设$g(a)=M(a)-N(a)$.\\\\\n(1) 求$g(a)$的函数解析式;\\\\\n(2) 求函数$g(a)$的最大值与最小值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326720,13 +327164,15 @@ "id": "020626", "content": "请研究函数$f(x)=-x+\\dfrac cx$($c \\neq 0$)的性质(不需要证明), 并画出大致图像.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326739,13 +327185,15 @@ "id": "020627", "content": "函数$f(x)=\\dfrac{x^2+1}x$($x>0$)的最小值为\\blank{50}, 当$x=$\\blank{50}时取到.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326758,13 +327206,15 @@ "id": "020628", "content": "函数$f(x)=\\dfrac{x^2+x+1}x$的单调增区间为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326777,13 +327227,15 @@ "id": "020629", "content": "函数$f(x)=\\dfrac{x^2+x+1}{x+1}$的定义域为\\blank{50}, 值域为\\blank{50}, 图像的对称中心的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326796,13 +327248,15 @@ "id": "020630", "content": "函数$f(x)=\\dfrac{x^2+x+1}{x^2+1}$的值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326815,13 +327269,15 @@ "id": "020631", "content": "函数$y=\\dfrac{x^2+5}{\\sqrt {x^2+4}}$的奇偶性为\\blank{50}, 值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326834,13 +327290,15 @@ "id": "020632", "content": "函数$f(x)=\\dfrac x{x^2+1}$, $x \\in[-5,5]$的单调减区间为\\blank{50}, 值域为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326853,13 +327311,15 @@ "id": "020633", "content": "求函数$f(x)=\\dfrac{x+1}{x^2+1}$, $x \\in[1,5]$的值域.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326872,13 +327332,15 @@ "id": "020634", "content": "已知函数$f(x)=\\dfrac 1{x^2+1}$.\\\\\n(1) 写出定义域、值域;\\\\\n(2) 判断奇偶性, 并说明理由;\\\\\n(3) 写出单调区间, 并利用二次函数、反比例函数的单调性证明你的结论;\\\\\n(4) 作出大致图像.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326891,13 +327353,15 @@ "id": "020635", "content": "研究函数$f(x)=\\dfrac{x+1}{x^2-x+1}$.\\\\\n(1) 写出定义域、值域;\\\\\n(2) 判断奇偶性, 并说明理由;\\\\\n(3) 写出单调区间;\\\\\n(4) 对于(3)的结论请选择一个单调区间进行证明.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326910,13 +327374,15 @@ "id": "020636", "content": "已知常数$a, b \\in \\mathbf{R}$, 函数$f(x)=\\dfrac{a x}{x^2+b}$.\\\\ \n(1) 写出定义域;\\\\\n(2) 当$b=-1$时, 若函数$y=f(x)$在区间$(-1,1)$上是严格增函数, 求$a$的取值范围;\\\\\n(3) 求$a, b$满足什么条件时, 函数存在最大值.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326929,13 +327395,15 @@ "id": "020637", "content": "试用解析式将圆的面积$S$表示成圆的周长$C$的函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326948,13 +327416,15 @@ "id": "020638", "content": "一个矩形的对角线长为$12$厘米, 试用解析式将它的一条边长$y$(厘米)表示成与这条边相邻的另一条边长$x$(厘米)的函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326967,13 +327437,15 @@ "id": "020639", "content": "已知上海到北京火车行驶路程为$1318$千米, 高速火车以每小时$300$千米的速度, 由上海开往北京. 试用解析式将行进中的火车到北京的路程$s$(千米)表示成行驶的时间$t$(时)的函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -326986,13 +327458,15 @@ "id": "020640", "content": "某中学的高一学生进行野外生存训练, 从甲地步行到乙地. 已知甲乙两地相距 $32$千米, 在前$3$小时内学生们每小时走$4$千米, 随后以每小时$5$千米的速度一直走到乙地. 设他们离开甲地的距离为$s$(千米)时, 所用的时间为$t$(时), 试用解析式将$s$(千米)表示成$t$(时)的函数.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327005,13 +327479,15 @@ "id": "020641", "content": "某商场对顾客实行购物优惠活动, 规定一次购物总额: \\textcircled{1} 如果不超过$500$元, 那么不予优惠; \\textcircled{2} 如果超过$500$元但不超过$1000$元, 那么按标价给予$9$折优惠; \\textcircled{3} 如果超过$1000$元, 那么其中的 $1000$元按\\textcircled{2}给予优惠, 超过$1000$元的部分给予$7$折优惠.\\\\\n设一次购物总额为$x$元, 优惠后实际付款额为$y$元, 试写出用$x$(元)表示$y$(元)的函数关系式.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327024,13 +327500,15 @@ "id": "020642", "content": "已知动点$P$从边长为$1$的正方形$ABCD$顶点$A$开始沿边界绕一圈, 若用$x$表示点$P$从$A$出发后的行程, $y$表示$PA$的长, 求$y$关于$x$的函数解析式.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\draw (2,2) node [above right] {$C$} coordinate (C);\n\\draw (0,2) node [above left] {$D$} coordinate (D);\n\\draw (2,1.2) node [right] {$P$} coordinate (P);\n\\draw (A) rectangle (C) (A) -- (P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327043,13 +327521,15 @@ "id": "020643", "content": "$\\triangle ABC$是边长为$1$的正三角形, $AD$为$BC$边上的高, 动点$P$由顶点$A$出发, 按逆时针方向在$\\triangle ABC$边界上移动一周, 设点$P$所移动的路程为$x$, 点$P$到$AD$的距离$PQ=y$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 2]\n\\draw (0,0) node [below left] {$A$} coordinate (A);\n\\draw (1,0) node [below right] {$B$} coordinate (B);\n\\draw (60:1) node [above] {$C$} coordinate (C);\n\\draw ($(B)!0.5!(C)$) node [above right] {$D$} coordinate (D);\n\\draw ($(A)!0.7!(B)$) node [below] {$P$} coordinate (P);\n\\draw ($(A)!(P)!(D)$) node [above left] {$Q$} coordinate (Q);\n\\draw (A) -- (B) -- (C) -- cycle (A) -- (D) (P) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 求$y$关于$x$的函数, 并画出函数的图像;\\\\\n(2) 当点$P$所移动路程为$\\dfrac 43$时, 点$P$到$AD$的距离为多少? 当点$P$到$AD$的距离为$\\dfrac 14$时, 点$P$所移动的路程为多少?", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327062,13 +327542,15 @@ "id": "020644", "content": "已知常数$a \\in \\mathbf{R}$, 关于$x$的方程$2 x^2+4 a x+3 a-1=0$有两个负根(包括重根), 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327081,13 +327563,15 @@ "id": "020645", "content": "已知常数$a \\in \\mathbf{R}$, 关于$x$的方程$a x^2-2 a x+1=0$至少有一个正根, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327100,13 +327584,15 @@ "id": "020646", "content": "已知常数$m \\in \\mathbf{R}$, 关于$x$的二次方程$2 x^2+3 x-5 m=0$有两个小于$1$的实根(包括重根), 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327119,13 +327605,15 @@ "id": "020647", "content": "已知常数$a \\in \\mathbf{R}$, 关于$x$的方程$x^2+(2 a-1) x+4-2 a=0$在$(-\\infty, 2)$与$(2,+\\infty)$上各有一个根, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327138,13 +327626,15 @@ "id": "020648", "content": "已知常数$a \\in \\mathbf{R}$, 关于$x$的方程$2 a x^2+2 x-a-3=0$在$[-1,1]$上有且只有一个根(重根算作多个根), 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327157,13 +327647,15 @@ "id": "020649", "content": "已知常数$a \\in \\mathbf{R}$, 关于$x$的方程$x^2+2 a x+2 a+1=0$.\\\\\n(1) 若方程有两根, 其中一根在$(-1,0)$上, 另一根在$(1,2)$上, 求$a$的取值范围;\\\\\n(2) 若方程在$(0,1)$上有两根(包括重根), 求$a$的取值范围;\\\\\n(3) 若方程在$(0,1)$上有且只有一个根(重根算作多个根), 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327176,13 +327668,15 @@ "id": "020650", "content": "函数$f(x)=x+3-\\dfrac 1x$的零点是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327195,13 +327689,15 @@ "id": "020651", "content": "若函数$y=x^2-k|x|+1$有 2 个零点, 则$k$的取值范围为", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327214,13 +327710,15 @@ "id": "020652", "content": "已知方程$x^2-4|x|+5=m$有四个互不相等的实数根, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327233,13 +327731,15 @@ "id": "020653", "content": "若函数$f(x)=(a^2+4 a-5) x^2-4(a-1) x+3$的图像都在$x$轴的上方(且与$x$轴无公共点), 则实数$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327252,13 +327752,15 @@ "id": "020654", "content": "已知常数$a \\in \\mathbf{R}$, 函数$x^2-a x+2>0$, 对$x \\in[1,2]$有解, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327271,13 +327773,15 @@ "id": "020655", "content": "已知$a$为实常数, 若关于$x$的不等式$|x^2+2 a x+3 a|\\leq 2$仅有一个实数解, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327290,13 +327794,15 @@ "id": "020656", "content": "若集合$A=\\{x||x^2-2 x+1+a |=a^2-6,\\ x \\in \\mathbf{R}\\}$中恰有两个元素, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327309,13 +327815,15 @@ "id": "020657", "content": "方程$\\dfrac{x^4+x^2+1}{x^4+x^2}=\\dfrac{201}{200}$是否有整数解? 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327328,13 +327836,15 @@ "id": "020658", "content": "用函数的观点在区间$(0,+\\infty)$上解不等式$x+\\dfrac 1x<\\dfrac{10}3$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327347,13 +327857,15 @@ "id": "020659", "content": "利用函数与不等式的关系, 在$a<0$时, 求解实系数一元二次不等式$a x^2+b x+c \\leq 0$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327366,13 +327878,15 @@ "id": "020660", "content": "关于$x$的方程$x^2+(m+1) x-2 m+1=0$在$[-1,1]$有解, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327385,13 +327899,15 @@ "id": "020661", "content": "已知函数$f(x)=x^2+10 x-a+3$, 当$x \\in[-2,+\\infty)$时, $f(x) \\geq 0$恒成立, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327404,13 +327920,15 @@ "id": "020662", "content": "已知关于$x$的方程$x^2-2 x=\\dfrac{3 a-2}{5-a}$在$(\\dfrac 12, 2)$上有实数根, 求实数$a$的取值围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327423,13 +327941,15 @@ "id": "020663", "content": "对于任意定义在区间$D$上的函数$f(x)$, 若实数$x_0 \\in D$, 满足$f(x_0)=x_0$, 就称$x_0$为函数$f(x)$在$D$上的一个不动点.\\\\\n(1) 求函数$f(x)=2 x+\\dfrac 1x-2$在$(0,+\\infty)$上的不动点;\\\\\n(2) 若函数$f(x)=2 x+\\dfrac 1x+a$在$(0,+\\infty)$上没有不动点, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327442,13 +327962,15 @@ "id": "020664", "content": "用函数观点可解得不等式$\\log_2(x+\\dfrac 1x)>2$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327461,13 +327983,15 @@ "id": "020665", "content": "方程$|a^x-1|=2 a$($a>0$, $a \\neq 1$)有两个不相等的实数解, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327480,13 +328004,15 @@ "id": "020666", "content": "若函数$k \\cdot 4^x-k \\cdot 2^{x+1}-4(k+5)=0$($k$为实常数)在区间$[0, 2]$上有解, 则$k$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327499,13 +328025,15 @@ "id": "020667", "content": "已知函数$f(x)=\\begin{cases}\\log_2 x, & x>0, \\\\3^x, & x \\leq 0,\\end{cases}$ 且函数$F(x)=f(x)+x-a$有且仅有两个零点, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327518,13 +328046,15 @@ "id": "020668", "content": "用函数观点解方程: $3^x+x^3+1=5$.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327537,13 +328067,15 @@ "id": "020669", "content": "已知函数$f(x)=\\lg (k x^2+2 x+1)$.\\\\ \n(1) 若$f(x)$定义域为$\\mathbf{R}$, 求$k$的取值范围;\\\\\n(2) 若$f(x)$值域为$\\mathbf{R}$, 求$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327556,13 +328088,15 @@ "id": "020670", "content": "用函数的观点求解下列问题: 已知关于$x$的方程$k \\cdot 9^x-k \\cdot 3^{x+1}-6(k+5)=0$在$[0,2]$上恰有一解, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327575,13 +328109,15 @@ "id": "020671", "content": "用函数观点解得不等式$2^x+\\log_2 x>2$的解集为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327594,13 +328130,15 @@ "id": "020672", "content": "设$a$为实数, 讨论关于$x$的方程$\\lg (x-1)+\\lg (3-x)=\\lg (a-x)$解的情况.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327613,13 +328151,15 @@ "id": "020673", "content": "已知函数$f(x)=2^x-\\dfrac 1{2^x}$. 若$2^x f(2 x)+m f(x) \\geq 0$对于$x \\in[1,2]$恒成立, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327632,13 +328172,15 @@ "id": "020674", "content": "函数$y=(x-1)^3+27$的零点为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327651,13 +328193,15 @@ "id": "020675", "content": "用二分法求出函数$y=x^3-2 x^2-x+1$在区间$(2,3)$内的一个零点.(精确到$0.1$)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327670,13 +328214,15 @@ "id": "020676", "content": "函数$y=\\sqrt [3]{x^3+2 x-3}$在区间$(0,+\\infty)$有\\blank{50}个零点.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327689,13 +328235,15 @@ "id": "020677", "content": "已知函数$y=2 x^3-3 x^2-18 x+28$在区间$(1,2)$上有且仅有一个零点. 用二分法求出该零点的近似值.(结果精确到$0.1$)", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327708,13 +328256,15 @@ "id": "020678", "content": "方程$\\dfrac 35 x+\\log_3 x=3$实数解可在的区间是\\bracket{20}.\n\\fourch{$(2,3)$}{$(3,4)$}{$(4,5)$}{$(5,6)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327727,13 +328277,15 @@ "id": "020679", "content": "函数$y=-x^2+(m-1) x+m^2-m+2$在下列哪个区间内一定有零点\\bracket{20}.\n\\fourch{$(-\\infty, 0]$}{$(0,1]$}{$(1,+\\infty)$}{$[3,+\\infty)$}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327746,13 +328298,15 @@ "id": "020680", "content": "若连续函数$y=f(x)$($x \\in \\mathbf{R}$)满足: $f(a) f(b)<0$, 则$f(x)$在区间$(a, b)$上\\bracket{20}.\n\\fourch{有且只有一个零点}{至少有一个零点}{至多有一个零点}{可能没有零点}", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327765,13 +328319,15 @@ "id": "020681", "content": "若$a0$, 那么是否该函数在区间$(a, b)$上一定无零点? 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327803,13 +328361,15 @@ "id": "020683", "content": "已知函数$f(x)=-x^2+\\dfrac 1{x-1}-1$, $x \\in(1,+\\infty)$.\\\\\n(1) 请用二分法求出函数$f(x)$的一个零点(结果精确到$0.1$)\\\\\n(2) 证明函数$f(x)$有且只有一个零点. 若存在请用二分法求出此零点(精确到$0.1$), 若不存在请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "2025届高一校本作业必修第五章试题", + "origin": "2025届高一校本作业必修第五章", "edit": [ "20221231\t王伟叶" ], @@ -327822,7 +328382,9 @@ "id": "020684", "content": "下列数列中, 为等差数列的是\\blank{50}.\\\\\n\\textcircled{1} $1,11,121$, \\textcircled{2} $1,2,1$, \\textcircled{3} $\\lg 2, \\lg 4, \\lg 8$, \\textcircled{4} $2, 2 ,2$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327841,7 +328403,9 @@ "id": "020685", "content": "已知数列$\\{a_n\\}$是等差数列, 请在下表中填入适当的数:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline $a_1$&$a_2$&$a_3$& 公差$d$&$a_5$\\\\\n\\hline $-3$& \\blank{20} & 6 & \\blank{20} & \\blank{20} \\\\\n\\hline \\blank{20}& $-5$ & \\blank{20} & 2 & \\blank{20} \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327860,7 +328424,9 @@ "id": "020686", "content": "已知数列$\\{a_n\\}$是等差数列, 请填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline$a_1$&$d$&$n$&$a_n$\\\\\n\\hline 5 & 10 & 12 & \\blank{20} \\\\\n\\hline$-5$& 6 & \\blank{20} & 61 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327879,7 +328445,9 @@ "id": "020687", "content": "等差数列$\\{a_n\\}$中,\\\\\n(1) 已知$a_1+a_6=12$, $a_4=7$, 则其通项公式$a_n=$\\blank{50};\\\\\n(2) 已知$a_7=2$, $a_8=-4$, 则$a_1=$\\blank{50}, $a_{10}=$\\blank{50};\\\\\n(3) 已知$a_4=2$, $a_8=26$, 则$a_{14}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327898,7 +328466,9 @@ "id": "020688", "content": "$\\dfrac{8-\\sqrt 2}2$与$\\dfrac{8+\\sqrt 2}2$的等差中项是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327917,7 +328487,9 @@ "id": "020689", "content": "$(a+b)^2$与$(a-b)^2$的等差中项是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327936,7 +328508,9 @@ "id": "020690", "content": "若$\\lg x, \\lg (3 x-2), \\lg (3 x+2)$成等差数列, 则$x=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327955,7 +328529,9 @@ "id": "020691", "content": "已知数列$\\{a_n\\}$是等差数列, 且$a_6=4, a_{14}=64$. 设$a_6$与$a_{14}$的等差中项为$x$, $a_6$与$x$的等差中项为$y$, $x$与$a_{14}$的等差中项为$z$, 求$x+y+z$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -327974,7 +328550,9 @@ "id": "020692", "content": "若等差数列$\\{a_n\\}$的前三项依次为$a-1, a+1,2 a+3$, 则其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -327993,7 +328571,9 @@ "id": "020693", "content": "在$-1$和$7$之间插入三个数, 使这五个数顺次成等差数列, 则插入的三个数依次是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328012,7 +328592,9 @@ "id": "020694", "content": "已知某个直角三角形的三边长成等差数列. 若这个三角形的斜边长为$5$, 则它的面积等于\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328031,7 +328613,9 @@ "id": "020695", "content": "已知三个数成等差数列, 首末两项之积为中间项的$5$倍, 后两项的和为第一项的$8$倍, 求这三个数.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328050,7 +328634,9 @@ "id": "020696", "content": "夏季高山上的温度从山脚起每升高$100$米降低$0.7^{\\circ} \\text{C}$. 已知山脚的温度是$34.9^{\\circ} \\text{C}$, 山顶的温度是$23^{\\circ} \\text{C}$, 求山的相对高度.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328069,7 +328655,9 @@ "id": "020697", "content": "在等差数列$\\{a_n\\}$中,\\\\\n(1) 若$a_1+a_2=30$, $a_3+a_4=60$, 则$a_5+a_6=$\\blank{50}, $a_7+a_8=$\\blank{50};\\\\\n(2) 若$a_3$、$a_{10}$是方程$x^2-3 x-5=0$的两根, 则$a_6+a_7=$\\blank{50};\\\\\n(3) 若$a_{15}=8$, $a_{60}=20$, 则$a_{75}=$\\blank{50};\\\\\n(4) 若$a_{m+n}=A$, $a_{m-n}=B$, 且$m>n$, $m$、$n$均为正整数, 则$a_m=$\\blank{50}. (用$A$、$B$表示)", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328088,7 +328676,9 @@ "id": "020698", "content": "已知等差数列$\\{a_n\\}$的公差为$3$. 若$a_3+a_5+a_7=39$, 则$a_{2022}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328107,7 +328697,9 @@ "id": "020699", "content": "在等差数列$\\{a_n\\}$中, $a_1=\\dfrac 18$, 若恰好从第$8$项开始有$a_n>1$($n \\in \\mathbf{N}$, $n \\geq 8$), 则公差$d$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328126,7 +328718,9 @@ "id": "020700", "content": "由公差为$d$的等差数列$a_1, a_2, a_3, \\cdots$, 重新组成的数列$a_1+a_3, a_2+a_4, a_3+a_5, \\cdots, a_n+a_{n+2}, \\cdots$\\bracket{20}.\n\\twoch{是以$d$为公差的等差数列}{是以$2 d$为公差的等差数列}{是以$3 d$公差的等差数列}{不是等差数列}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -328145,7 +328739,9 @@ "id": "020701", "content": "若等差数列$\\{a_n\\}$的公差$d=2$, $a_2+a_5+a_8+\\cdots+a_{26}=20$, 求$a_1+a_4+a_7+\\cdots+a_{25}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328164,7 +328760,9 @@ "id": "020702", "content": "已知$a, b, c, d$成等差数列, 求证:$2 a-3 b, 2 b-3 c, 2 c-3 d$成等差数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328183,7 +328781,9 @@ "id": "020703", "content": "在等差数列$\\{a_n\\}$中,\\\\\n(1) 若$a_1+a_2+a_3=-24$, $a_{17}+a_{18}+a_{19}=72$, 则$a_{10}=$\\blank{50};\\\\\n(2) 若$a_7=p$, $a_{14}=q$($p \\neq q$), 则$a_{21}=$\\blank{50};\\\\\n(3) 若$a_3+a_4+a_5+a_6+a_7=450$, 则$a_2+a_8=$\\blank{50};\\\\\n(4) 若$a_1+a_4+a_7=39$, $a_2+a_5+a_8=33$, 则$a_3+a_6+a_9=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328202,7 +328802,9 @@ "id": "020704", "content": "首项为$-24$的等差数列从第$10$项开始为正数, 则公差$d$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328221,7 +328823,9 @@ "id": "020705", "content": "公差$d$($d \\neq 0$)的等差数列$\\{a_n\\}$中, 若$a_1, a_2$为关于$x$的方程$x^2-a_3 x+a_4=0$的两根, 则$\\{a_n\\}$的通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328240,7 +328844,9 @@ "id": "020706", "content": "在递增的等差数列$\\{a_n\\}$中, 已知$a_3+a_6+a_9=12$, $a_3 a_6 a_9=28$, 则通项$a_n$等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328259,7 +328865,9 @@ "id": "020707", "content": "已知等差数列$\\{a_n\\}$分别满足下列条件, 求解相应问题.\\\\\n(1) $d=\\dfrac 13$, $n=37$, $S_n=629$, 则$a_1=$\\blank{50};\\\\\n(2) $d=2$, $n=15$, $a_n=-10$, 则$S_n=$\\blank{50};\\\\\n(3) $a_1=20$, $a_n=54$, $S_n=999$, 则$d=$\\blank{50};\\\\\n(4) $a_1=\\dfrac 56$, $d=-\\dfrac 16$, $S_n=-5$, 则$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328278,7 +328886,9 @@ "id": "020708", "content": "在等差数列$1,4,7, \\cdots, 100$中, 项数$n=$\\blank{50}, 其前$n$项和$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328297,7 +328907,9 @@ "id": "020709", "content": "在数列$\\{a_n\\}$中, 若$a_1=5$, $a_{n+1}=a_n-2$, 则其前$n$项和$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328316,7 +328928,9 @@ "id": "020710", "content": "等差数列$\\{a_n\\}$中,\\\\ \n(1) 若$a_4+a_6=6$, 若其前$5$项和$S_5=10$, 则其公差$d=$\\blank{50};\\\\\n(2) 若其前$15$项和为$135$, 则其第$8$项$a_8=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328335,7 +328949,9 @@ "id": "020711", "content": "已知数列$\\{a_n\\}$的前$n$项和为$S_n=n^2+n+1$, 则其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328354,7 +328970,9 @@ "id": "020712", "content": "求$1000$以内能被$7$整除的所有正整数的和.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328373,7 +328991,9 @@ "id": "020713", "content": "已知等差数列$\\{a_n\\}$的前$5$项和为$0$, 前$10$项和为$-100$, 求这个数列的前$20$项和. 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328392,7 +329012,9 @@ "id": "020714", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=47-2 n$, 当其前$n$项和$S_n$最大值时, $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328411,7 +329033,9 @@ "id": "020715", "content": "等差数列$\\{a_n\\}$中,\\\\ \n(1) 若$a_1=\\lg 3$, $a_2=\\lg 6$, 且前$5$项和$S_5=\\lg x$, 则$x=$\\blank{50};\\\\\n(2) 若$S_4=1$, $S_8=4$则$a_{17}+a_{18}+a_{19}+a_{20}=$\\blank{50};\\\\\n(3) 若$S_{10}=100$, $S_{100}=10$, 则$S_{110}=$\\blank{50};\\\\\n(4) 若公差$d=\\dfrac 12$, 且$a_1+a_3+a_5+\\cdots+a_{97}+a_{99}=60$, 则$S_{100}=$\\blank{50}.\n(5) 若$S_m=S_n=l$($m \\neq n$), 则$a_1+a_{m+n}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328430,7 +329054,9 @@ "id": "020716", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=n a_n-n^2+n$, 求$a_{100}-a_{99}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328449,7 +329075,9 @@ "id": "020717", "content": "已知数列$\\{a_n\\}$的前$n$项和$S_n=2 n^2-3 n-1$, 求其通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328468,7 +329096,9 @@ "id": "020718", "content": "已知$\\{a_n\\}$是等差数列, 记$b_n=\\dfrac{a_1+a_2+\\cdots+a_n}n$, 求证: 数列$\\{b_n\\}$是等差数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328487,7 +329117,9 @@ "id": "020719", "content": "在等差数列中, 若前$n$项和$S_n=3 n^2+4 n$, 则公差$d=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328506,7 +329138,9 @@ "id": "020720", "content": "等差数列$\\{a_n\\}$中, $S_n$是其前$n$项和, 公差为$d$.\\\\\n(1) 若$a_2+a_5+a_9+a_{13}+a_{16}=20$, 则$S_{17}=$\\blank{50};\\\\\n(2) 若$a_1+a_2+a_3=-24$, $a_{16}+a_{17}+a_{18}=1$, 则$S_{18}=$\\blank{50};\\\\\n(3) 若$3 a_4=7 a_7$, 且$a_1>0$, 则当$S_n$取得最大值时, $n=$\\blank{50};\\\\\n(4) 若$a_1+a_2+a_3+a_4+a_5=30$, $a_6+a_7+a_8+a_9+a_{10}=80$, \n则$a_{11}+a_{12}+a_{13}+a_{14}+a_{15}=$\\blank{50};\\\\\n(5) 若$d=1$, $S_{98}=137$, 则$a_2+a_4+a_6+\\cdots+a_{98}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328525,7 +329159,9 @@ "id": "020721", "content": "等差数列$\\{a_n\\}$中, $S_n$是其前$n$项和, 公差为$d$, $S_{20}=10M$, 则下列结论中正确的是\\bracket{20}.\n\\fourch{$M=2 a_5+a_{10}$}{$M=a_1+2 a_{10}$}{$M=a_5+a_{15}$}{$M=2 a_{10}+d$}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -328544,7 +329180,9 @@ "id": "020722", "content": "若数列$\\{a_n\\}$的前$n$项和为$S_n=n^2+3 n+c$, 则它的通项公式$a_n=$\\blank{50}; 当\\blank{50}时, $\\{a_n\\}$是等差数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328563,7 +329201,9 @@ "id": "020723", "content": "已知等差数列$\\{a_n\\}$中, $\\dfrac{a_6}{a_4}=\\dfrac 7{11}$, $S_n$是此数列的前$n$项和, 求$\\dfrac{S_{11}}{S_7}$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328582,7 +329222,9 @@ "id": "020724", "content": "已知两等差数列$\\{a_n\\}$、$\\{b_n\\}$前$n$项和分别为$A_n$、$B_n$, 若$\\dfrac{A_n}{B_n}=\\dfrac{4 n+3}{2 n+5}$, 求$a_8: b_8$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328601,7 +329243,9 @@ "id": "020725", "content": "设等差数列$\\{a_n\\}$前$n$项和为$S_n$, 已知$a_3=11$, $S_{15}>0$, $S_{16}<0$.\\\\\n(1) 求公差$d$的取值范围;\\\\\n(2) $S_n$取最大值时的$n$的值.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328620,7 +329264,9 @@ "id": "020726", "content": "若一个等差数列共有$2 n+1$项, 其中奇数项之和为$290$, 偶数项之和为 $261$, 则第$n+1$项为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328639,7 +329285,9 @@ "id": "020727", "content": "一个有限项的等差数列, 它的前$6$项和为$48$, 后$6$项和为$132$, 所有项和为$255$, 则该数列共有\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328658,7 +329306,9 @@ "id": "020728", "content": "若$\\{a_n\\}$是等差数列, 数列$\\{b_n\\}$满足$b_n=(\\dfrac 12)^{a_n}$, $b_1+b_2+b_3=\\dfrac{21}8$, $b_1 b_2 b_3=\\dfrac 18$, 求通项公式$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328677,7 +329327,9 @@ "id": "020729", "content": "若数列$\\{a_n\\}$的前$n$项和$S_n=n^2-4 n+2$, 记$T_n=|a_1|+|a_2|+\\cdots+|a_{10}|$, 求$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -328696,7 +329348,9 @@ "id": "020730", "content": "已知数列$\\{a_n\\}$是等比数列, 请在下表中填入适当的数:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline$a_1$&$a_2$&$a_3$& 公比$q$&$a_5$\\\\\n\\hline \\blank{20}&$-1$& 3 & \\blank{20}&\\blank{20} \\\\\n\\hline \\blank{20}& 4 &\\blank{20} & 2 & \\blank{20}\\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328715,7 +329369,9 @@ "id": "020731", "content": "已知$\\{a_n\\}$是等比数列, 根据所给的条件填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline$a_1$&$q$&$n$&$a_n$\\\\\n\\hline 9 & \\blank{20} & 4 & 243 \\\\\n\\hline \\blank{20} &$-2$& 7 & 32 \\\\\n\\hline 4 &$\\dfrac 12$&\\blank{20} &$\\dfrac 1{32}$\\\\\n\\hline 3 &$\\dfrac 23$& 3 & \\blank{20}\\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328734,7 +329390,9 @@ "id": "020732", "content": "已知数列$\\{a_n\\}$是等比数列, 且$a_9=-2, a_{12}=-16$, 则这个数列的通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328753,7 +329411,9 @@ "id": "020733", "content": "在等比数列$\\{a_n\\}$中, 如果公比$q<0$, $a_1+a_2=30$, $a_3+a_4=120$, 那么这个数列的通项公式为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328772,7 +329432,9 @@ "id": "020734", "content": "在等比数列$\\{a_n\\}$中, 若$a_1=1$, $a_3=4$, 则$a_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328791,7 +329453,9 @@ "id": "020735", "content": "在等比数列$\\{a_n\\}$中, 若$a_1=1$, $a_5=4$, 则$a_3=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328810,7 +329474,9 @@ "id": "020736", "content": "在等比数列$\\{b_n\\}$中, $b_4=3$, 则该数列前七项之积是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328829,7 +329495,9 @@ "id": "020737", "content": "已知直角三角形的斜边长为$c$, 两条直角边长分别为$a$、$b$, 且$a, b, c$成等比数列, 则$a: c$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328848,7 +329516,9 @@ "id": "020738", "content": "在$2, x, 8, y$四个数中, 前三个数成等比数列, 后三个数成等差数列, 则$x y=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328867,7 +329537,9 @@ "id": "020739", "content": "在等比数列$\\{a_n\\}$中, 若$a_1=1$, $a_{10}=5$, 则$a_2 \\cdot a_3 \\cdot a_4 \\cdot a_5 \\cdot a_6 \\cdot a_7 \\cdot a_8 \\cdot a_9=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -328886,7 +329558,9 @@ "id": "020740", "content": "已知$\\{a_n\\}$是等比数列. 下列命题中, 不正确的是\\bracket{20}.\n\\onech{若$a_n$恒正, 则$\\{\\lg a_n\\}$是等差数列}{若$a_n$恒正, 则$\\dfrac{a_1+a_{n+2}}2 \\geq \\sqrt {a_2 a_{n+1}}$}{$a_{n+1}$一定是$a_n$与$a_{n+2}$的等比中项}{$a_{n-r}$与$a_{n+r}$($r0$, 求$a+a^3+a^5+\\cdots+a^{2 n-1}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329247,7 +329957,9 @@ "id": "020759", "content": "已知等比数列$\\{a_n\\}$的前$10$项和为$10$, 前$30$项和为$70$, 求这个数列的前$20$项和.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329266,7 +329978,9 @@ "id": "020760", "content": "在数列$\\{a_n\\}$中, 若$a_1=2$, 且$a_n=S_{n-1}$($n \\geq 2$, $n \\in \\mathbf{N}$), 则数列$\\{a_n\\}$的前$n$项和$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329285,7 +329999,9 @@ "id": "020761", "content": "在等差数列$\\{a_n\\}$中, 若$a_{10}=0$, 则有等式$a_1+a_2+\\cdots+a_n=a_1+a_2+\\cdots+a_{19-n}(n<19)$成立. 类比上述性质, 相应地, 在等比数列$\\{b_n\\}$中, 若$b_9=1$, 则有等式\\blank{50}成立.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329304,7 +330020,9 @@ "id": "020762", "content": "若数列$\\{a_n\\}$的前$n$项和为$S_n=\\dfrac 1{2^n}(3^n-2^n)$, 则数列$\\{a_n\\}$\\blank{20}等差数列, \\blank{20}等比数列(填``是''``不是'').", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329323,7 +330041,9 @@ "id": "020763", "content": "若数列$\\{a_n\\}$的前$n$项和$S_n=2^n-1$, 则其前$n$项的平方和$a_1^2+a_2^2+\\cdots+a_n^2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329342,7 +330062,9 @@ "id": "020764", "content": "$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{\\dfrac 12+\\dfrac 14+\\cdots+\\dfrac 1{2^n}}{\\dfrac 13-\\dfrac 19+\\cdots+(-1)^{n+1} \\dfrac 1{3^n}}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329361,7 +330083,9 @@ "id": "020765", "content": "无穷等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 则数列$\\{a_n\\}$有极限是数列$\\{S_n\\}$有极限的\\bracket{20}条件.\n\\fourch{充分不必要}{必要不充分}{充要}{即不充分又不必要}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -329380,7 +330104,9 @@ "id": "020766", "content": "$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{6-2+4-8+\\cdots+(-2)^{n+1}}{4+3+9+27+\\cdots+3^n}=$\\bracket{20}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -329399,7 +330125,9 @@ "id": "020767", "content": "若$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{3^n}{3^{n+1}+(a+1)^n}=\\dfrac 13$, 则实数$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329418,7 +330146,9 @@ "id": "020768", "content": "设无穷等比数列$\\{a_n\\}$所有奇数项和为$36$, 偶数项和为$12$, 则$q=$\\blank{50}, $a_1=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329437,7 +330167,9 @@ "id": "020769", "content": "若首项为$a_1$, 公比为$q$的无穷等比数列$\\{a_n\\}$的前$n$项和总小于这个数列的各项和, 则首项$a_1$、公比$q$的一组取值可以是$(a_1, q)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329456,7 +330188,9 @@ "id": "020770", "content": "已知等比数列$\\{a_n\\}$中, $q \\in \\mathbf{R}$, 且$a_1+a_2+a_3=24$, $a_4+a_5+a_6=-3$, 则$\\displaystyle\\lim _{n \\to \\infty}(a_1+a_2+\\cdots+a_n)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329475,7 +330209,9 @@ "id": "020771", "content": "如右图, 正方形上连接等腰直角三角形, 直角三角形边上再连接正方形, $\\cdots \\cdots$, 无限重复, 若第一个正方形边长为$2$, 则所有这些正方形和三角形的面积总和为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (1,0) node [below] {$2$};\n\\draw (0,0) rectangle (2,2);\n\\draw (0,2) --++ (1,1) --++ (1,-1);\n\\draw (0,2) --++ (-1,1) coordinate (A) --++ (1,1) --++ (1,-1);\n\\draw (A) --++ (0,1) --++ (1,0);\n\\draw (A) --++ (-1,0) --++ (0,1) --++ (1,0);\n\\draw (A) ++ (-1,0) node [below left] {$\\cdots$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329494,7 +330230,9 @@ "id": "020772", "content": "设无穷等比数列$\\{a_n\\}$满足$\\displaystyle\\lim _{n \\to \\infty}(a_1+a_3+a_5+\\cdots+a_{2 n-1})=\\dfrac 83$, 求首项$a_1$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329513,7 +330251,9 @@ "id": "020773", "content": "数列$\\{a_n\\}$的前$n$项和为$S_n$, 若$a_n=5S_n-3$, $n$是正整数, 求$\\displaystyle\\lim _{n \\to \\infty}(a_1+a_3+a_5+\\cdots+a_{2 n-1})$", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329532,7 +330272,9 @@ "id": "020774", "content": "设$\\{a_n\\}$为无穷等比数列, $\\{a_n\\}$中每一项都是它后面所有项和的$4$ 倍, 且$a_5=\\dfrac{16}{625}$, 求它的所有偶数项之和.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329551,7 +330293,9 @@ "id": "020775", "content": "等差数列$\\{a_n\\}$、$\\{b_n\\}$的前$n$项和分别为$S_n$、$T_n$, 若$\\dfrac{S_n}{T_n}=\\dfrac{2 n}{3 n+1}$, 则$\\displaystyle\\lim _{n \\to \\infty} \\dfrac{a_n}{b_n}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329570,7 +330314,9 @@ "id": "020776", "content": "若$\\log_2(9 x)+\\log_2(x-\\dfrac 13)=1$, 则$\\displaystyle\\lim _{n \\to \\infty}(1+x+x^2+\\cdots+x^n)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329589,7 +330335,9 @@ "id": "020777", "content": "设等比数列$\\{a_n\\}$的前$n$项和为$S_n$, 已知$S_5=1$, $S_{10}=\\dfrac 32$, 且$\\displaystyle\\lim _{n \\to \\infty} S_n$存在, 则$\\displaystyle\\lim _{n \\to \\infty} S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329608,7 +330356,9 @@ "id": "020778", "content": "一个无穷递缩等比数列中, 所以奇数项的和比所有偶数项的和大$27$, 又这个无穷递缩等比数列所有项的和为$135$, 求这个数列的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329627,7 +330377,9 @@ "id": "020779", "content": "设$\\{a_n\\}$是首项为$a$, 公比为$q$($q>0$)的等比数列, 前$n$项和为$S_n$, 若$G_n=a_1^2+a_2^2+\\cdots+a_n^2$($n \\in N$), 求$\\displaystyle\\lim_{n \\to \\infty} \\frac{S_n}{G_n}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329646,7 +330398,9 @@ "id": "020780", "content": "已知数列$\\{a_n\\}$的通项公式为$a_n=2(1+3 n)$, 填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline$n$& 1 & 2 & 3 &$\\cdots$& 11 &$\\cdots$&\\blank{20}&$\\cdots$&\\blank{20} \\\\\n\\hline$a_n$&\\blank{20} &\\blank{20}&\\blank{20} &$\\cdots$& \\blank{20}&$\\cdots$& 128 &$\\cdots$& 602 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329665,7 +330419,9 @@ "id": "020781", "content": "$a_n=\\dfrac 1n+\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\cdots+\\dfrac 1{n^2}$中每个$a_n$有\\blank{50}个加数项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329684,7 +330440,9 @@ "id": "020782", "content": "在数列$\\{a_n\\}$中, 若$a_n=2^n+3$($n$是正整数), 则它的前$4$项分别是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329703,7 +330461,9 @@ "id": "020783", "content": "在数列$\\{a_n\\}$中, 若$a_n=3 n+1$, 则$2017$是这数列的第\\blank{50}项.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329722,7 +330482,9 @@ "id": "020784", "content": "若数列$\\{a_n\\}$的通项公式为$a_n=(n^2-7 n+10) \\cdot 0.7^n$, 则该数列中共有\\blank{50}项是负数, 它们是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329741,7 +330503,9 @@ "id": "020785", "content": "写出无穷数列$\\{a_n\\}$的一个通项公式, 使它的前$4$项分别是下列各数:\\\\\n(1) $4,8,12,16$;\\\\\n(2) $\\dfrac 12, \\dfrac 23, \\dfrac 34, \\dfrac 45$;\\\\\n(3) $-\\dfrac 1{2 \\times 1}, \\dfrac 1{2 \\times 2},-\\dfrac 1{2 \\times 3}, \\dfrac 1{2 \\times 4}$;\\\\\n(4) $1,-\\sqrt[3]2, \\sqrt[3]3,-\\sqrt[3]4$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329760,7 +330524,9 @@ "id": "020786", "content": "根据下列条件, 分别写出数列$\\{a_n\\}$的第$3$项至第$5$项.\\\\\n(1) $a_1=1$, $a_n=2 a_{n-1}+3$;\\\\\n(2) $a_1=1$, $a_n=2 a_{n-1}+2^n$;\\\\\n(3) $a_1=1$, $a_n=a_{n-1}+\\dfrac 1{a_{n-1}}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329779,7 +330545,9 @@ "id": "020787", "content": "若点$(n, a_n)$在一次函数$y=\\dfrac 23 x+\\dfrac 53$的图像上, 则数列$\\{a_n\\}$的通项公式是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329798,7 +330566,9 @@ "id": "020788", "content": "数列$1,0,-1,0,1,0,-1,0, \\cdots$的一个可能的通项公式是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329817,7 +330587,9 @@ "id": "020789", "content": "在数列$\\{a_n\\}$中, 若$a_1=1$, $a_2=4$, 且$a_{n+2}=a_{n+1}-a_n$, 则$a_{2021}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329836,7 +330608,9 @@ "id": "020790", "content": "下面四个结论中, 正确的是\\blank{50}.\\\\\n\\textcircled{1} 数列中的符号$\\{a_n\\}$表示:$a_1, a_2, \\cdots, a_n$;\\\\\n\\textcircled{2} 每个数列都有通项公式;\\\\\n\\textcircled{3} 数列的通项公式的表达形式可以不唯一;\\\\\n\\textcircled{4} 数列可以看作是定义在正整数集上的函数.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329855,7 +330629,9 @@ "id": "020791", "content": "写出下列数列$\\{a_n\\}$的一个通项公式.\\\\\n(1) $1,3,7,15,31, \\cdots$:\\blank{50};\\\\\n(2) $\\dfrac 12, \\dfrac 45, \\dfrac 9{10}, \\dfrac{16}{17}, \\dfrac{25}{26}, \\cdots$:\\blank{50};\\\\\n(3) $-\\dfrac 12, \\dfrac 12,-\\dfrac 12, \\dfrac 12,-\\dfrac 12, \\cdots$:\\blank{50};\\\\\n(4) $0,1,0,1,0,1, \\cdots$:\\blank{50};\\\\\n(5) $\\dfrac 23,-\\dfrac 38, \\dfrac 4{15},-\\dfrac 5{24}, \\cdots$:\\blank{50};\\\\\n(6) $9,99,999,9999,99999, \\cdots$:\\blank{50};\\\\", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329874,7 +330650,9 @@ "id": "020792", "content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}=a_n+n+1$, 则$a_6=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329893,7 +330671,9 @@ "id": "020793", "content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_2=6$, $a_{n+2}=-a_n$.\\\\\n(1) 写出这个数列的前$8$项;\\\\\n(2) 根据第(1)题的结论, 猜想这个数列的项所具有的特征, 并给予证明.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -329912,7 +330692,9 @@ "id": "020794", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=a_n+\\dfrac 1{2^n}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329931,7 +330713,9 @@ "id": "020795", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=\\dfrac{2(n+1)}n a_n$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329950,7 +330734,9 @@ "id": "020796", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=\\dfrac 23 a_n+1$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329969,7 +330755,9 @@ "id": "020797", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=\\dfrac{a_n}{2 a_n+1}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -329988,7 +330776,9 @@ "id": "020798", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=a_n+2 n+5$, 其通项$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330007,7 +330797,9 @@ "id": "020799", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_{n+1}=5^n a_n$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330026,7 +330818,9 @@ "id": "020800", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $S_{n+1}=4 a_n+2$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330045,7 +330839,9 @@ "id": "020801", "content": "已知数列$\\{a_n\\}$, $a_1=2$, $a_{n+1}=1-\\dfrac 1{a_n}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330064,7 +330860,9 @@ "id": "020802", "content": "已知数列$\\{a_n\\}$, 前$n$项和$S_n=n^2+n+1$, 求其通项公式$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330083,7 +330881,9 @@ "id": "020803", "content": "已知数列$\\{a_n\\}$, $a_1=1$, $a_2=-1$, $a_{n+1}-2 a_n+a_{n-1}=0$($n \\geq 2$), 求其通项公式$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330102,7 +330902,9 @@ "id": "020804", "content": "已知数列$\\{a_n\\}$, $a_1=-1$, $a_{n+1}=2 a_n+4 \\cdot 3^{n-1}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330121,7 +330923,9 @@ "id": "020805", "content": "已知数列$\\{a_n\\}$, $a_1=2$, $a_{n+1}=\\dfrac{2^{n+1} a_n}{a_n+2^{n+1}}$, 其通项公式$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330140,7 +330944,9 @@ "id": "020806", "content": "数列$\\{a_n\\}$中, $a_1=1$, $n \\geq 2$时, $a_n, S_n, S_n-\\dfrac 12$成等比数列, 求数列$\\{a_n\\}$通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330159,7 +330965,9 @@ "id": "020807", "content": "在正数列$\\{a_n\\}$中, $a_1=1$, 前$n$项和为$S_n$, 且$2S_{n-1}=\\dfrac 1{a_n}-a_n$($n \\geq 2$).\\\\ \n(1) 求证: 数列$\\{S_n^2\\}$是等差数列;\\\\\n(2) 求其通项公式$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330178,7 +330986,9 @@ "id": "020808", "content": "数列$\\{a_n\\}$的前$n$项和$S_n=n^2-7 n-8$.\\\\\n(1) 求$\\{a_n\\}$的通项公式;\\\\\n(2) 求$\\{|a_n|\\}$的前$n$项和$T_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330197,7 +331007,9 @@ "id": "020809", "content": "用数学归纳法证明: ``$2+3+4+\\cdots+n=\\dfrac{(n-1)(n+2)}2$($n$为正整数)''时, 第一步是取$n=$\\blank{50}验证.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330216,7 +331028,9 @@ "id": "020810", "content": "设$n$为正整数, 用数学归纳法证明: $1+2+3+\\cdots+(n+4)=\\dfrac 12(n+4)(n+5)$时, 当$n=1$时, 左边应为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330235,7 +331049,9 @@ "id": "020811", "content": "用数学归纳法证明:\n$1+a+a^2+\\cdots+a^{n+1}=\\dfrac{1-a^{n+2}}{1-a}$($a \\neq 1$, $n$为正整数). 在验证$n=1$时, 等式左边为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330254,7 +331070,9 @@ "id": "020812", "content": "用数学归纳法证明:\n$(n+1)(n+2) \\cdots(n+n)=2^n \\cdot 1 \\cdot 3 \\cdot \\cdots \\cdot(2 n-1)$($n$为正整数), 等式左边从$k$到$k+1$时, 需增加的代数式是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330273,7 +331091,9 @@ "id": "020813", "content": "用数学归纳法证明: $1 \\times 2+2 \\times 5+\\cdots+n(3 n-1)=n^2(n+1)$($n$为正整数).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330292,7 +331112,9 @@ "id": "020814", "content": "若$f(n)=\\dfrac 1{n+1}+\\dfrac 1{n+2}+\\dfrac 1{n+3}+\\cdots+\\dfrac 1{2 n}$($n$为正整数), 则$f(k+1)=f(k)+$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330311,7 +331133,9 @@ "id": "020815", "content": "如果$f(n)=1+\\dfrac 12+\\dfrac 13+\\cdots+\\dfrac 1n$($n$为正整数), 则$f(2^{k+1})-f(2^k)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330330,7 +331154,9 @@ "id": "020816", "content": "用数学归纳法证明:\n$1-2^2+3^2-4^2+\\cdots+(-1)^{n-1} n^2=(-1)^{n-1} \\dfrac{n(n+1)}2$($n$为正整数).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330349,7 +331175,9 @@ "id": "020817", "content": "用数学归纳法证明:\n``$1+\\dfrac 1{2^2}+\\dfrac 1{3^2}+\\cdots+\\dfrac 1{n^2}<\\dfrac{2 n-1}n$($n>1$, $n \\in \\mathbf{N}$)''的过程中, 第一步是验证不等式\\blank{50}成立.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330368,7 +331196,9 @@ "id": "020818", "content": "不等式$2^n>n^2$($n$为正整数)成立的条件是\\blank{50}, 因此若要用数学归纳法证明 ``这个不等式, 在证明过程中的第一步中, $n$可取的第一个值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330387,7 +331217,9 @@ "id": "020819", "content": "在数列$\\{a_n\\}$中, 已知$a_1=1$, $a_2=2$. 若$a_n=2 a_{n-1}-a_{n-2}$($n \\geq 3$且$n \\in \\mathbf{N}$), 则$a_3=$\\blank{50}, $a_4=$\\blank{50}, $a_5=$\\blank{50}, 进而猜想$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330406,7 +331238,9 @@ "id": "020820", "content": "已知数列$\\{a_n\\}$满足$(a_{n+1}-a_n)^2-2(a_{n+1}+a_n)+1=0$, 且$a_1=1$, $a_{n+1} \\geq a_n$($n$为正整数). 则$a_2=$\\blank{50}, $a_3=$\\blank{50}, 进而猜想$a_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330425,7 +331259,9 @@ "id": "020821", "content": "观察下列数字:\n\\begin{center}\n\\begin{tabular}{ccccccc}\n1 \\\\\n2 & 3 & 4\\\\\n3 & 4 & 5 & 6 & 7 \\\\\n4 & 5 & 6 & 7 & 8 & 9 & 10\\\\\n\\end{tabular}\\\\\n$\\cdots\\cdots\\cdots$\n\\end{center}\n猜想第$n$行的各数之和$S_n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330444,7 +331280,9 @@ "id": "020822", "content": "用数学归纳法证明:$1^3+2^3+3^3+\\cdots+n^3=[\\dfrac 12 n(n+1)]^2$($n$为正整数).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330463,7 +331301,9 @@ "id": "020823", "content": "已知数列$\\{a_n\\}$满足: $a_1=1$, $a_{n+1}=\\dfrac{3 a_n}{a_n+3}$, $a_n \\neq 0$($n$为正整数).\\\\\n(1) 求$a_2$、$a_3$、$a_4$;\\\\\n(2) 猜想$\\{a_n\\}$的通项公式, 并用数学归纳法加以证明.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330482,7 +331322,9 @@ "id": "020824", "content": "对于数列$1,3,5, \\cdots,(2 n-1), \\cdots$, 按如下规律分组: $1,(3,5),(7,9,11),(13,15,17,19), \\cdots$即依次取$1$项, $2$项, $3$项, $\\cdots$各为一组, 则第$n$组的各数之和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330501,7 +331343,9 @@ "id": "020825", "content": "已知$f(x)=2 x+1$, $f^{(1)}(x)=f(x)$, $f^{(n)}(x)=f[f^{(n-1)}(x)]$($n \\geq 2$, $n \\in \\mathbf{N}$), 猜想$f^{(n)}(x)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330520,7 +331364,9 @@ "id": "020826", "content": "是否存在常数$a$、$b$、$c$, 使等式$1 \\cdot(n^2-1^2)+2 \\cdot(n^2-2^2)+\\cdots+n \\cdot(n^2-n^2)=$$a n^4+b n^2+c$对一切正整数$n$都成立? 证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330539,7 +331385,9 @@ "id": "020827", "content": "已知数列$\\{a_n\\}$满足$a_1=\\dfrac 12$, $a_1+a_2+\\cdots+a_n=n^2 a_n$($n$为正整数), 试用数学归纳法证明: $a_n=\\dfrac 1{n(n+1)}$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330558,7 +331406,9 @@ "id": "020828", "content": "已知数列$\\{a_n\\}$满足$a_1=1$, 设该数列的前$n$项和为$S_n$, 且$S_n, S_{n+1}, 2 a_1$成等差数列, 用数学归纳法证明:$S_n=\\dfrac{2^n-1}{2^{n-1}}$($n$为正整数).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330577,7 +331427,9 @@ "id": "020829", "content": "用数学归纳法证明: $2^{3 n}-1$($n$为正整数)能被$7$整除.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330596,7 +331448,9 @@ "id": "020830", "content": "已知$a>0$, $x_1=\\sqrt a$, $x_{n+1}=\\sqrt {a+x_n}$($n \\in \\mathbf{N}$, $n\\ge 1$), 求证: $\\{x_n\\}$是严格增数列.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330615,7 +331469,9 @@ "id": "020831", "content": "已知数列$\\{a_n\\}$的通项为$a_n=\\dfrac{8 n}{(2 n-1)^2(2 n+1)^2}$.\n设$S_n$为$\\{a_n\\}$的前$n$项和. 计算$S_1$、$S_2$、$S_3$、$S_4$, 试由此猜想计算$S_n$的公式, 并用数学归纳法加以证明.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330634,7 +331490,9 @@ "id": "020832", "content": "求证: $1+\\dfrac{1}{\\sqrt{2}}+\\dfrac{1}{\\sqrt{3}}+\\cdots+\\dfrac{1}{\\sqrt{n}}>\\sqrt{n}$($n \\in \\mathbf{N}$, $n \\geq 2$).", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330653,7 +331511,9 @@ "id": "020833", "content": "是否存在大于$1$的正整数$m$, 使得$f(n)=(2 n+7) \\cdot 3^n+9$对任意正整数$n$都能被$m$整除? 若存在, 求出$m$的最大值, 并证明你的结论; 若不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330672,7 +331532,9 @@ "id": "020834", "content": "数列$\\{a_n\\}$中, $a_1=0$, $a_{n+1}=a_n+n-1$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330691,7 +331553,9 @@ "id": "020835", "content": "数列$\\{a_n\\}$中, $a_1=0$, $a_{n+1}=a_n+3^{n-1}$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330710,7 +331574,9 @@ "id": "020836", "content": "设数列$\\{a_n\\}$满足$a_1=1$, $a_{n+1}=3^{n-1} \\cdot a_n$, 求通项公式$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330729,7 +331595,9 @@ "id": "020837", "content": "设数列$\\{a_n\\}$满足$a_1=1$且$a_{n+1}=a_n+\\dfrac 1{n(n+1)}$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330748,7 +331616,9 @@ "id": "020838", "content": "设数列$\\{a_n\\}$满足$a_1=1$且$a_{n+1}=2 a_n+2$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330767,7 +331637,9 @@ "id": "020839", "content": "设数列$\\{a_n\\}$满足$a_1=2$且$a_{n+1}=2 a_n+2^n$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330786,7 +331658,9 @@ "id": "020840", "content": "设数列$\\{a_n\\}$满足$a_1=1$且$a_{n+1}=(1+\\dfrac 1n) \\cdot a_n$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330805,7 +331679,9 @@ "id": "020841", "content": "设数列$\\{a_n\\}$满足$a_1=2$, $a_{n+1}+a_n=2 a_{n+1} \\cdot a_n$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330824,7 +331700,9 @@ "id": "020842", "content": "设数列$\\{a_n\\}$满足$a_1=1$, $a_2=2$且$a_{n+2}=(1-\\dfrac 2n) \\cdot a_n$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330843,7 +331721,9 @@ "id": "020843", "content": "设数列$\\{a_n\\}$满足$a_1=2$, $a_{n+1}=a_n^2$, 求通项公式$a_n$.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330862,7 +331742,9 @@ "id": "020844", "content": "设数列$\\{a_n\\}$满足$a_1=1$且$a_{n+1} \\cdot a_n=2^n$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330881,7 +331763,9 @@ "id": "020845", "content": "数列$\\{a_n\\}$中前$n$项和$S_n$, 满足$S_n=2 a_n-n$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330900,7 +331784,9 @@ "id": "020846", "content": "数列$\\{a_n\\}$中前$n$项和$S_n$, 满足$S_n=2 a_n-n^2$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330919,7 +331805,9 @@ "id": "020847", "content": "数列$\\{a_n\\}$的前$n$项和为$S_n$. 若$a_1=2$, $S_n=4 a_{n+1}+2$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330938,7 +331826,9 @@ "id": "020848", "content": "设数列$\\{a_n\\}$满足$a_1=2$, $a_2=1$, $a_{n+2}=4 a_{n+1}-4 a_n$, 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330957,7 +331847,9 @@ "id": "020849", "content": "设数列$\\{a_n\\}$满足$a_1=1$, $a_2=2$, $a_{n+2}=\\sqrt {a_{n+1} a_n}$.\\\\\n(1) 求证: 对于任意的$n$为正整数, $a_n>0$;\\\\\n(2) 求数列$\\{a_n\\}$的通项公式.", "objs": [], - "tags": [], + "tags": [ + "第四单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -330976,7 +331868,9 @@ "id": "020850", "content": "试用集合符号表示:\\\\\n(1) 点$A$在直线$l$上, 点$B$不在直线$l$上: \\blank{100};\\\\\n(2) 点$A$在平面$\\alpha$内, 点$B$不在平面$\\beta$内: \\blank{100}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -330995,7 +331889,9 @@ "id": "020851", "content": "点$P$在直线$l$上, $l$在平面$\\alpha$内, $P$、$l, \\alpha$之间的关系可用集合符号表示为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331014,7 +331910,9 @@ "id": "020852", "content": "用集合语言表示下列语句并画图表示:\\\\\n(1) 点$M$是平面$\\alpha$与平面$\\beta$的一个公共点;\\\\\n(2) 平面$\\alpha$与平面$\\beta$没有公共点, 且直线$l$与平面$\\alpha$和平面$\\beta$分别交于点$A$和点$B$;\\\\\n(3) 平面$\\alpha$与平面$\\beta$交于直线$l$, 且直线$l$与平面$\\gamma$没有公共点;\\\\\n(4) 直线$l$在平面$\\alpha$上, 点$M$在平面$\\alpha$上, 但不在直线$l$上;\\\\\n(5) 平面$\\alpha$与平面$\\beta$交于直线$l$. 直线$a$与平面$\\alpha$、平面$\\beta$都没有公共点.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331033,7 +331931,9 @@ "id": "020853", "content": "判断下列说法是否正确? 并说明理由.\\\\\n(1) 平面的形状是平行四边形;\\\\\n(2) 任何一个平面图形都是一个平面;\\\\\n(3) 一个平面长$3$米, 宽$4$米, 其面积等于$12$平方米;\\\\\n(4) 如果梯形$ABCD$中的两条边在平面$\\alpha$内, 那么它的另外两条边也在平面$\\alpha$内.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331052,7 +331952,9 @@ "id": "020854", "content": "用集合语言表示下列语句并画图:\\\\\n如果平面$\\alpha$与平面$\\beta$交于直线$l$, 平面$\\alpha$与平面$\\gamma$交于直线$m$, 平面$\\beta$与平面$\\gamma$交于直线$n$, 且直线$l$与直线$m$平行, 那么直线$l$、$m$、$n$两两平行.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331071,7 +331973,9 @@ "id": "020855", "content": "下面四个命题的叙述: (其中$A$、$B$表示不同的点, $a$表示直线, $\\alpha$表示平面)\\\\\n\\textcircled{1} 因为$A \\in \\alpha$, $B \\in \\alpha$, 所以$AB \\subset \\alpha$;\\\\\n\\textcircled{2} 因为$A \\in \\alpha$, $B \\in \\alpha$, 所以$AB \\in \\alpha$;\\\\\n\\textcircled{3} 因为$A \\notin a$, $a$不在$\\alpha$内, 所以$A \\notin \\alpha$;\\\\\n\\textcircled{4} 若$l$与平面$\\alpha$平行, $P$在直线$l$上, 则$P$不在平面$\\alpha$内.\\\\\n其中叙述方法和推理过程都正确的命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331090,7 +331994,9 @@ "id": "020856", "content": "已知点$A$、$B$、$C$、$D$在平面$\\alpha$上, 求证: 四边形$ABCD$的四条边所在的直线都在平面$\\alpha$上.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331109,7 +332015,9 @@ "id": "020857", "content": "两个不重合的平面最多把空间分割成几个部分? 并画图表示.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331128,7 +332036,9 @@ "id": "020858", "content": "三个不重合的平面最多把空间分割成几个部分? 并画图表示.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331147,7 +332057,9 @@ "id": "020859", "content": "三点能确定一个平面的条件是\\blank{50}, 两条直线能确定一个平面的条件是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331166,7 +332078,9 @@ "id": "020860", "content": "有一个公共点的两个平面相交于\\blank{50}的一条直线.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331185,7 +332099,9 @@ "id": "020861", "content": "三条直线两两相交, 由这三条直线所确定的平面的个数是\\bracket{20}.\n\\fourch{$1$}{$2$}{$3$}{$1$或$3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331204,7 +332120,9 @@ "id": "020862", "content": "若空间中有四个点, 则``这四个点中有三点在同一直线上''是``这四个点在同一平面上''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331223,7 +332141,9 @@ "id": "020863", "content": "若$A \\in \\alpha$, $B \\in \\alpha$, $C \\in AB$, 其中$A$、$B$、$C$是不同的三点, $\\alpha$是平面, 则下列关系错误的是\\bracket{20}.\n\\fourch{$C \\in \\alpha$}{$C \\notin \\alpha$}{$AB \\subset \\alpha$}{$AC \\subset \\alpha$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331242,7 +332162,9 @@ "id": "020864", "content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 如果一条直线与另两条直线都相交, 那么这三条直线必共面;\\\\\n(2) 如果三条直线相互平行, 那么这三条直线在同一个平面上;\\\\\n(3) 如果两个平面有无数个公共点, 那么这两个平面重合.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331261,7 +332183,9 @@ "id": "020865", "content": "请你指出下列命题中, 与命题``直线$a$上两点$A$、$B$在平面$\\alpha$内''不等价的命题:\\blank{50}.\\\\\n\\textcircled{1} $a \\subset \\alpha$;\\\\\n\\textcircled{2} 直线$a$上只有$A$、$B$两点在$\\alpha$内;\\\\\n\\textcircled{3} 平面$\\alpha$经过$a$;\\\\\n\\textcircled{4} 直线上所有点都在$\\alpha$内.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331280,7 +332204,9 @@ "id": "020866", "content": "如图, 已知$A \\in l$, $B \\in l$, $O \\notin l$, 求证: 直线$OA$、$OB$、$OC$在同一平面上.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,0) node [left] {$O$} coordinate (O);\n\\draw (-2,2) node [above] {$A$} coordinate (A);\n\\draw (-0.5,2) node [above right] {$B$} coordinate (B);\n\\draw (1.5,2) node [above] {$C$} coordinate (C);\n\\draw (-2.5,2) -- (2,2) node [right] {$l$};\n\\draw ($(O)!-0.2!(A)$) -- ($(O)!1.2!(A)$);\n\\draw ($(O)!-0.2!(B)$) -- ($(O)!1.2!(B)$);\n\\draw ($(O)!-0.2!(C)$) -- ($(O)!1.2!(C)$);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331299,7 +332225,9 @@ "id": "020867", "content": "过空间任意一点引四条直线, 最多可以确定几个平面? 为什么?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331318,7 +332246,9 @@ "id": "020868", "content": "已知$A$、$B$、$C$、$D$、$E$是空间的五个点, 且线段$CE$、$AC$和$BD$两两相交且不共点, 求证:$A$、$B$、$C$、$D$、$E$这五个点在同一平面上.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331337,7 +332267,9 @@ "id": "020869", "content": "给出以下关于斜二测直观图的结论, 其中正确的是\\blank{50}.\\\\\n\\textcircled{1} 角的水平放置的直观图一定是角;\\\\\n\\textcircled{2} 相等的角在直观图中仍相等;\\\\\n\\textcircled{3} 相等的线段在直观图中仍然相等;\\\\\n\\textcircled{4} 若两条线段平行, 则在直观图中对应的两条线段仍然平行.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331356,7 +332288,9 @@ "id": "020870", "content": "利用斜二测画法得到: \\textcircled{1} 三角形的直观图是三角形; \\textcircled{2} 平行四边形的直观图是平行四边形; \\textcircled{3} 正方形的直观图是正方形 ; \\textcircled{4} 菱形的直观图是菱形. 以上说法正确的是\\bracket{20}.\n\\fourch{\\textcircled{1}}{\\textcircled{1}\\textcircled{2}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{2}\\textcircled{3}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331375,7 +332309,9 @@ "id": "020871", "content": "如图为一平面图形的直观图, 则此平面图形可能是选项中的\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, y = {(45:0.5cm)}]\n\\draw [->] (-0.5,0) -- (1.5,0) node [below] {$x'$};\n\\draw [->] (0,-0.5) -- (0,2) node [left] {$y'$};\n\\draw (0,0) node [below right] {$O'$};\n\\draw (1,0) -- (1,1) -- ({-sqrt(2)/4},1) -- (0,0) -- cycle; \n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw (1,0) -- (1,1) -- (0,1) -- (0,0) -- cycle; \n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw (1,0) -- (1.5,1) -- (0,1) -- (0,0) -- cycle; \n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw (1,0) -- (1,1) -- ({-sqrt(2)/4},1) -- (0,0) -- cycle; \n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw (1,0) -- (1.5,1) -- (0.5,1) -- (0,0) -- cycle; \n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331394,7 +332330,9 @@ "id": "020872", "content": "如图所示, $\\triangle A' B' C'$是水平放置的$\\triangle ABC$的直观图, 则在$\\triangle ABC$的三边及中线$AD$中, 最长的线段是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, y = {(45:0.5cm)}]\n\\draw [->] (-0.5,0) -- (3,0) node [below] {$x'$};\n\\draw [->] (0,-0.5) -- (0,3) node [left] {$y'$};\n\\draw (0,0) node [below] {$O'(B')$};\n\\draw (0,2) node [left] {$A'$} coordinate (A');\n\\draw (1,0) node [below] {$D'$} coordinate (D');\n\\draw (2,0) node [below] {$C'$} coordinate (C'); \n\\draw (A') -- (D') (A') -- (C');\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$AB$}{$AD$}{$BC$}{$AC$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331413,7 +332351,9 @@ "id": "020873", "content": "已知$\\triangle ABC$是边长为$2 a$的正三角形, 那么它的平面直观图$\\triangle A'B'C'$的面积为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt 3}2 a^2$}{$\\dfrac{\\sqrt 3}4 a^2$}{$\\dfrac{\\sqrt 6}4 a^2$}{$\\sqrt 6 a^2$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331432,7 +332372,9 @@ "id": "020874", "content": "用斜二测画法画长、宽、高分别为$10$、$8$、$6$的长方体的直观图.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331451,7 +332393,9 @@ "id": "020875", "content": "如图, 水平放置的$\\triangle ABC$的斜二测直观图是图中的$\\triangle A'B'C'$, 已知$A'C'=6$, $B'C'=6$, 则$AB$边的实际长度是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, y = {(45:0.5cm)}]\n\\draw [->] (-1,0) -- (2,0) node [below] {$x'$};\n\\draw [->] (0,-2) -- (0,4) node [left] {$y'$};\n\\draw (0,0) node [below] {$O'$} node [above] {$C'$};\n\\draw (1.5,0) node [below] {$A'$} -- (0,3) node [left] {$B'$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331470,7 +332414,9 @@ "id": "020876", "content": "如图, 是$\\triangle AOB$用斜二测画法画出的直观图, 则$\\triangle AOB$的面积是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, y = {(45:0.5cm)}, scale = 0.5]\n\\draw [->] (-1,0) -- (6,0) node [below] {$x'$};\n\\draw [->] (0,-2) -- (0,12) node [left] {$y'$};\n\\draw (0,0) node [below] {$O'$};\n\\draw (4,0) node [below] {$4$} node [above right] {$B'$} -- (2,8) node [right] {$A'$} -- (0,0);\n\\draw (2,0.4) -- (2,0) node [below] {$2$};\n\\draw (0.2,4) -- (0,4) node [left] {$2$} (0.2,8) -- (0,8) node [left] {$4$};\n\\draw [dashed] (2,0) -- (2,8) -- (0,8);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331489,7 +332435,9 @@ "id": "020877", "content": "已知一个正方形的直观图是一个平行四边形, 其中有一边长为$4$, 则此正方形的面积是\\bracket{20}.\n\\fourch{$16$}{$64$}{$16$或$64$}{无法确定}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331508,7 +332456,9 @@ "id": "020878", "content": "在下列选项中, 利用斜二测画法, 边长为$1$的正三角形$ABC$的直观图不是全等三角形的一组是\\bracket{20}.\n\\twoch{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (0.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (0,0) node [below left] {$(B)$} --++ (120:1) node [above] {$C$} --++ (240:1) node [below] {$A$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) node [below right] {$(A)$} --++ (60:1) node [above] {$C$} --++ (-60:1) node [below] {$B$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1,0) -- (1,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-0.5,0) node [below left] {$A$} --++ (60:1) node [above right] {$C$} --++ (-60:1) node [below] {$B$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) node [below right] {$(A)$} --++ (60:1) node [above] {$C$} --++ (-60:1) node [below] {$B$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) node [below right] {$(A)$} --++ (60:1) node [above] {$C$} --++ (-60:1) node [below] {$B$};\n\\end{tikzpicture}\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1) -- (0,1) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0.5) node [left] {$A$} --++ (-30:1) node [below] {$C$} --++ (-150:1) node [left] {$B$};\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) node [below right] {$(A)$} --++ (60:1) node [above] {$C$} --++ (-60:1) node [below] {$B$};\n\\end{tikzpicture}\\begin{tikzpicture}[>=latex]\n\\draw [->] (-0.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-1.5) -- (0,0.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) node [above right] {$(A)$} --++ (-60:1) node [below] {$C$} --++ (60:1) node [above] {$B$};\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331527,7 +332477,9 @@ "id": "020879", "content": "如图所示, 四边形$ABCD$是一个梯形, $CD\\parallel AB$, $CD=AO=1$, 三角形$AOD$为等腰直角三角形, $O$为$AB$的中点, 试求梯形$ABCD$水平放置的直观图的面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-1.5,0) -- (1.5,0) node [below] {$x$};\n\\draw [->] (0,-0.5) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (-1,0) node [below] {$A$} -- (0,1) node [left] {$D$} -- (1,1) node [above right] {$C$} -- (1,0) node [below] {$B$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331546,7 +332498,9 @@ "id": "020880", "content": "设$\\angle A$与$\\angle B$两边分别平行, $\\angle A=30^{\\circ}$, 则$\\angle B=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331565,7 +332519,9 @@ "id": "020881", "content": "正方体$ABCD-A_1B_1C_1D_1$中, $M$、$N$分别为棱$CD$、$AD$中点, 则直线$MN$与$A_1C_1$的位置关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331584,7 +332540,9 @@ "id": "020882", "content": "直线$a\\parallel b, c$、$d$为不重合的直线, 且$a\\parallel c$, $b\\parallel d$, 则$c$与$d$的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331603,7 +332561,9 @@ "id": "020883", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $P$为$CC_1$上一点, 试过点$P$画棱$AD$的平行线, 并说明画法和理由.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{2}\n\\def\\n{1.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(C)!0.3!(C1)$) circle (0.03) node [right] {$P$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331622,7 +332582,9 @@ "id": "020884", "content": "在空间四边形$ABCD$中, $E$、$F$、$G$、$H$分别是$AB$、$BC$、$CD$、$DA$的中点, 且$AC=BD$, 试证明$EFGH$是平面图形, 并分析四边形$EFGH$的性质.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0) node [right] {$D$} coordinate (D);\n\\draw (1.3,3) node [above] {$A$} coordinate (A);\n\\draw (1.5,0.8) node [above right] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(B)$) node [left] {$E$} coordinate (E);\n\\draw ($(B)!0.5!(C)$) node [below] {$F$} coordinate (F);\n\\draw ($(C)!0.5!(D)$) node [below] {$G$} coordinate (G);\n\\draw ($(D)!0.5!(A)$) node [right] {$H$} coordinate (H);\n\\draw (A) -- (B) -- (D) -- cycle;\n\\draw [dashed] (A) -- (C) -- (B) (C) -- (D);\n\\draw [dashed] (E) -- (F) -- (G) -- (H) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331641,7 +332603,9 @@ "id": "020885", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $O_1$、$O_2$分别是正方形$ABB_1A_1$, 和正方形$DCC_1D_1$的对角线的交点, 求证:$\\angle A_1O_1D_1=\\angle CO_2B$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B1)$) node [right] {$O_1$} coordinate (O1);\n\\draw ($(C)!0.5!(D1)$) node [left] {$O_2$} coordinate (O2);\n\\draw (A1) -- (O1);\n\\draw [dashed] (D1) -- (O1) (B) -- (O2) (C) -- (O2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331660,7 +332624,9 @@ "id": "020886", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $P$、$Q$分别为$CC_1$、$AA_1$的中点, 求证: $BP\\parallel D_1Q$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331679,7 +332645,9 @@ "id": "020887", "content": "已知直线$l$上有$A$、$B$、$C$三点, 过这三点分别作三条互相平行$a$、$b$、$c$直线. 求证: $l$、$a$、$b$、$c$四条直线都在同一平面内.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331698,7 +332666,9 @@ "id": "020888", "content": "已知正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$、$G$、$H$分别是棱$AB$、$BC$、$C_1D_1$、$A_1D_1$的中点, 求证: 点$E$、$F$、$G$、$H$在同一平面内.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\filldraw ($(A)!0.5!(B)$) circle (0.03) node [below] {$E$} coordinate (E);\n\\filldraw ($(B)!0.5!(C)$) circle (0.03) node [below right] {$F$} coordinate (F);\n\\filldraw ($(C1)!0.5!(D1)$) circle (0.03) node [above] {$G$} coordinate (G);\n\\filldraw ($(A1)!0.5!(D1)$) circle (0.03) node [above left] {$H$} coordinate (H);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331717,7 +332687,9 @@ "id": "020889", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是棱$A_1B_1$、$DC$上的点, 且$EB_1=DF$, 求证:$\\triangle AED_1 \\cong \\triangle C_1FB$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331736,7 +332708,9 @@ "id": "020890", "content": "没有公共点的两条直线的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331755,7 +332729,9 @@ "id": "020891", "content": "对于空间中的两条直线$l_1$与$l_2$, 则``$l_1$与$l_2$异面''是``$l_1$与$l_2$无公共点''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331774,7 +332750,9 @@ "id": "020892", "content": "已知$a$、$b$是异面直线, $c\\parallel a$, 那么$c$与$b$\\bracket{20}.\n\\fourch{一定是异面直线}{一定是相交直线}{不可能是平行直线}{不可能是相交直线}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331793,7 +332771,9 @@ "id": "020893", "content": "分别和两条异面直线平行的两条直线的位置关系是\\bracket{20}.\n\\fourch{一定平行}{一定相交}{一定异面}{相交或异面}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331812,7 +332792,9 @@ "id": "020894", "content": "已知$a, b$是两条异面直线, 且分别在平面$\\alpha, \\beta$内. 若$\\alpha \\cap \\beta=l$, 则直线$l$必定\\bracket{20}.\n\\twoch{分别与$a, b$相交}{至少与$a, b$之一相交}{与$a, b$都不相交}{至多与$a, b$之一相交}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -331831,7 +332813,9 @@ "id": "020895", "content": "已知$A$、$B$、$C$、$D$是不共面的四个点, 证明: 直线$AB$与$CD$是异面直线.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331850,7 +332834,9 @@ "id": "020896", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $AC$与$DB$交于点$O$, $B_1O$与\n$AA_1$是不是异面直线? 为什么?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B)!0.5!(D)$) node [below] {$O$} coordinate (O);\n\\draw [dashed] (A) -- (C) (B) -- (D) (O) -- (B1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331869,7 +332855,9 @@ "id": "020897", "content": "正方体$12$条棱所在直线共能组成\\blank{50}对异面直线.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331888,7 +332876,9 @@ "id": "020898", "content": "已知点$P$在四边形$ABCD$所在平面外, 如果把两条异面直线看成一对, 那么$P$与四边形$ABCD$的四个顶点的连线以及此四边形的四条边所在的直线共$8$条直线中, 异面直线共有多少对?\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A) -- (2,0,0) node [below] {$B$} coordinate (B) -- (2,0,-2) node [above right] {$C$} coordinate (C);\n\\draw (1,2,-1) node [above] {$P$} coordinate (P);\n\\draw (0,0,-2) node [above right] {$D$} coordinate (D);\n\\draw (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw [dashed] (A) -- (D) -- (C) (D) -- (P);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331907,7 +332897,9 @@ "id": "020899", "content": "如图, 已知直线$c$、$d$分别与异面直线$a$、$b$相交于$E$、$F$和$G$、$H$四点, 求证: 直线$c$、$d$是异面直线.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) coordinate (L) -- (2,1) coordinate (R);\n\\draw [name path = lineU] (L) --++ (2.5,0) --++ (2,1) -- (R);\n\\draw [name path = lineD] (L) --++ (1,2) --++ (2,1) -- (R);\n\\filldraw ($(L)!0.3!(R)$) circle (0.03) node [left] {$E$} coordinate (E);\n\\filldraw ($(L)!0.5!(R)$) circle (0.03) node [above] {$F$} coordinate (F);\n\\filldraw (E) ++ (1.1,1.6) circle (0.03) node [left] {$G$} coordinate (G);\n\\filldraw (F) ++ (2.2,0.1) circle (0.03) node [above] {$H$} coordinate (H);\n\\draw (F) -- ($(F)!1.1!(H)$) node [midway, below] {$b$} (E) -- ($(E)!1.1!(G)$) node [midway, left] {$a$};\n\\path [name path = GH] ($(G)!-0.3!(H)$) -- ($(G)!1.3!(H)$) node [midway, above] {$d$};\n\\path (E) -- (F) node [midway, below] {$c$};\n\\path [name intersections = {of = GH and lineU, by = P}];\n\\path [name intersections = {of = GH and lineD, by = Q}];\n\\draw (G) -- (H) ($(P)!-0.1!(G)$) -- (P) ($(Q)!-0.1!(H)$) -- (Q);\n\\draw [dashed] (P) -- (H) (Q) -- (G);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331926,7 +332918,9 @@ "id": "020900", "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$中, 直线$BA_1$与$CC_1$所成角的大小为\\blank{50}, 直线$BA_1$与$B_1C$所成角的大小为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B) -- (A1) (B1) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -331945,7 +332939,9 @@ "id": "020901", "content": "正方体$ABCD-A'B'C'D'$中, $P$、$Q$分别为$A'B'$、$BB'$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A'$} coordinate (A');\n\\draw (A') ++ (\\l,0,0) node [below right] {$B'$} coordinate (B');\n\\draw (A') ++ (\\l,0,-\\l) node [right] {$C'$} coordinate (C');\n\\draw (A') ++ (0,0,-\\l) node [left] {$D'$'} coordinate (D');\n\\draw (A') -- (B') -- (C');\n\\draw [dashed] (A') -- (D') -- (C');\n\\draw (A') ++ (0,\\l,0) node [left] {$A$} coordinate (A);\n\\draw (B') ++ (0,\\l,0) node [right] {$B$} coordinate (B);\n\\draw (C') ++ (0,\\l,0) node [above right] {$C$} coordinate (C);\n\\draw (D') ++ (0,\\l,0) node [above left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A') -- (A) (B') -- (B) (C') -- (C);\n\\draw [dashed] (D') -- (D);\n\\draw ($(A')!0.5!(B')$) node [below] {$P$} coordinate (P);\n\\draw ($(B)!0.5!(B')$) node [right] {$Q$} coordinate (Q);\n\\draw (A) -- (P) (C) -- (Q) (B) -- (D);\n\\end{tikzpicture}\n\\end{center}\n(1) 求直线$AP$与$CQ$所成的角的大小;\\\\\n(2) 求直线$AP$与$BD$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331964,7 +332960,9 @@ "id": "020902", "content": "如图, 在棱长为$2$的正方体$ABCD-A'B'C'D'$中, $E$、$F$分别\n是$A' B'$和$AB$的中点, 求异面直线$A'F$与$CE$所成角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A'$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B'$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C'$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D'$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(B1)$) node [above] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(B)$) node [below] {$F$} coordinate (F);\n\\draw (A1) -- (F);\n\\draw [dashed] (C) -- (E);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -331983,7 +332981,9 @@ "id": "020903", "content": "如图, 点$A$是$BCD$所在平面外一点, $AD=BC$, $E,F$分别是$AB,CD$的中点, 且$EF=\\dfrac{\\sqrt 2}2AD$, 求异面直线$AD$和$BC$所成角.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [right] {$D$} coordinate (D);\n\\draw (D) ++ (-2,0,0) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1.5,0,1) node [below] {$C$} coordinate (C);\n\\draw (D) ++ (-0.4,{sqrt(1.5*1.5+1-0.4*0.4-0.6*0.6)},0.6) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [above] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [below] {$F$} coordinate (F);\n\\draw ($(A)!0.5!(C)$) node [right] {$G$} coordinate (G);\n\\draw (A) -- (B) -- (C) -- (D) --cycle (E) -- (G) -- (F) (A) -- (C);\n\\draw [dashed] (B) -- (D) (E) -- (F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332002,7 +333002,9 @@ "id": "020904", "content": "$A$是$\\triangle BCD$平面外的一点, $E$、$F$分别是$BC$、$AD$的中点.\\\\ \n(1) 求证: 直线$EF$与$BD$是异面直线;\\\\\n(2) 若$AC \\perp BD$, 且$AC=BD$, 求$EF$与$BD$所成的角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332021,7 +333023,9 @@ "id": "020905", "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中, $AA_1=AB=2, AD=1$, 点\n$E, F, G$分别是$DD_1, AB, CC_1$的中点, 则异面直线$A_1E$与$GF$所成的角是\n\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{1.5}\n\\def\\n{1.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(C)!0.5!(C1)$) node [right] {$G$} coordinate (G);\n\\draw ($(A)!0.5!(B)$) node [below] {$F$} coordinate (F);\n\\draw ($(D)!0.5!(D1)$) node [right] {$E$} coordinate (E);\n\\draw [dashed] (A1) -- (E) (F) -- (G);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\arccos \\dfrac{\\sqrt {15}}5$}{$\\dfrac{\\pi}4$}{$\\arccos \\dfrac{\\sqrt {10}}5$}{$\\dfrac{\\pi}2$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332040,7 +333044,9 @@ "id": "020906", "content": "下图为正方体平面展开图, 那么在正方体中, 有以下四个结论:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below left] {$E$} coordinate (E);\n\\draw (1,1) node [above left] {$D$} coordinate (D);\n\\draw (1,0) node [below left] {$A$} coordinate (A);\n\\draw (2,0) node [below right] {$B$} coordinate (B);\n\\draw (2,-1) node [below right] {$F$} coordinate (F);\n\\draw (2,1) node [above right] {$C$} coordinate (C);\n\\draw (3,1) node [above] {$M$} coordinate (M);\n\\draw (1,2) node [above left] {$N$} coordinate (N);\n\\draw (E) --++ (4,0) --++ (0,1) --++ (-4,0) -- cycle;\n\\draw (N) --++ (1,0) --++ (0,-3) --++ (-1,0) -- cycle;\n\\draw (M) --++ (0,-1);\n\\draw (E) -- (D) (N) -- (C) (M) -- (B) (A) -- (F);\n\\end{tikzpicture}\n\\end{center}\n\\textcircled{1} $BM$与$ED$平行;\n\\textcircled{2} $CN$与$BE$是异面直线;\n\\textcircled{3} $CN$与$BM$成$60^{\\circ}$;\n\\textcircled{4} $DN \\perp FN$.\n以上四个结论中, 正确结论的序号是\\blank{50}.( 写出所有情况)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332059,7 +333065,9 @@ "id": "020907", "content": "正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是$BD$、$B_1C$上的点, 且\n$BE=B_1F=\\dfrac 23B_1C$, 求直线$EF$与$CD$所成的角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(B1)!0.5!(C)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(D)$) node [below left] {$E$} coordinate (E);\n\\draw [dashed] (E) -- (F) (B) -- (D);\n\\draw (B1) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332078,7 +333086,9 @@ "id": "020908", "content": "判断下列命题的对错:\\\\\n\\blank{20}(1) 过直线$l$外一点$A$只能作一条直线与$l$平行;\\\\\n\\blank{20}(2) 过平面$\\alpha$外一点$A$只能作一条直线与$\\alpha$平行;\\\\\n\\blank{20}(3) 若两条直线$a$、$b$都与第三条直线$c$垂直, 则$a\\parallel b$;\\\\\n\\blank{20}(4) 若两条直线$a$、$b$都与第三条直线$c$平行, 则$a\\parallel b$;\\\\\n\\blank{20}(5) 若直线$l$在平面$\\alpha$外, 则$l$不可能与$\\alpha$内的无数条直线都相交;\\\\\n\\blank{20}(6) 若直线$l$与平面$\\alpha$不平行, 则$l$与$\\alpha$内的任何一条直线都不平行.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332097,7 +333107,9 @@ "id": "020909", "content": "正方体$ABCD-A_1B_1C_1D_1$中, 点$M, N$分别是正方形$A_1ADD_1$, $DCC_1D_1$的中心, 则直线$MN$平面$ABCD$的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332116,7 +333128,9 @@ "id": "020910", "content": "若平面$\\alpha$平面$\\beta=CD$, 平面$\\alpha \\cap$平面$\\gamma=EF$, 平面$\\beta \\cap$平面$\\gamma=AB$, $AB\\parallel$平面$\\alpha$, 则$CD$与$EF$的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332135,7 +333149,9 @@ "id": "020911", "content": "下列命题中不正确的命题是\\blank{50}.(写出所有编号)\\\\\n\\textcircled{1} 若直线$a \\subset \\alpha$, 直线$b \\subset \\beta, a\\parallel b$, 则$\\alpha$和$\\beta$没有公共点;\\\\\n\\textcircled{2} 若$a\\parallel$平面$\\beta$, 直线$a\\parallel \\alpha$, 直线$b \\subset \\beta$, 则$a\\parallel b$;\\\\\n\\textcircled{3} 若平面$\\alpha\\parallel$平面$\\beta$, 直线$a\\parallel \\alpha$, 直线$b \\subset \\beta$, 则$a$和$b$异面;\\\\\n\\textcircled{4} 若平面$\\alpha\\parallel$平面$\\beta$, 直线$a\\parallel \\alpha$, 则$a\\parallel \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332154,7 +333170,9 @@ "id": "020912", "content": "如图, 已知$M$、$N$、$P$、$Q$分别是空间四边形$ABCD$的边$AB$、$BC 、$\n$CD$、$DA$的中点. 求证:\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 1.5]\n\\draw (0,0,0) node [right] {$D$} coordinate (D);\n\\draw (D) ++ (-2,0,0) node [left] {$B$} coordinate (B);\n\\draw (B) ++ (1.5,0,1) node [below] {$C$} coordinate (C);\n\\draw (D) ++ (-0.4,{sqrt(1.5*1.5+1-0.4*0.4-0.6*0.6)},0.6) node [above] {$A$} coordinate (A);\n\\draw ($(A)!0.5!(B)$) node [left] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(C)$) node [below left] {$N$} coordinate (N);\n\\draw ($(C)!0.5!(D)$) node [below right] {$P$} coordinate (P);\n\\draw ($(D)!0.5!(A)$) node [right] {$Q$} coordinate (Q);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (M) -- (N) (A) -- (C) (P) -- (Q);\n\\draw [dashed] (B) -- (D) (N) -- (P) (M) -- (Q);\n\\end{tikzpicture}\n\\end{center}\n(1) 线段$MP$和$NQ$相交且互相平分;\\\\\n(2) 直线$AC\\parallel$平面$MNP$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332173,7 +333191,9 @@ "id": "020913", "content": "如图, 两个全等的正方形$ABCD$和$ABEF$所在的平面相交于直线$AB, M$、$N$分别是$AC$、$FB$的中点, 求证:$MN\\parallel$平面$BCE$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-150:0.7)}]\n\\draw (0,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,0) node [right] {$E$} coordinate (E);\n\\draw (2,2,0) node [right] {$F$} coordinate (F);\n\\draw (0,2,0) node [above] {$A$} coordinate (A);\n\\draw (0,0,2) node [below] {$C$} coordinate (C);\n\\draw (0,2,2) node [above] {$D$} coordinate (D);\n\\draw (C) -- (B) -- (E) -- (F) -- (A) -- (D) -- cycle (A) -- (B);\n\\draw ($(A)!0.5!(C)$) node [left] {$M$} coordinate (M);\n\\draw ($(B)!0.5!(F)$) node [right] {$N$} coordinate (N);\n\\draw (M) -- (N) (A) -- (C) (F) -- (B);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332192,7 +333212,9 @@ "id": "020914", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $P, Q$分别是$AD_1, BD$上的点, 且$AP=BQ$, 求证:$PQ\\parallel$平面$DCC_1D_1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.3!(D1)$) node [above] {$P$} coordinate (P);\n\\draw ($(B)!0.3!(D)$) node [above] {$Q$} coordinate (Q);\n\\draw [dashed] (A) -- (D1) (B) -- (D) (P) -- (Q);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332211,7 +333233,9 @@ "id": "020915", "content": "如图, $E,F$分别是空间四边形$ABCD$的边$BC$、$AD$的中点, 过$EF$且平行于$AB$的平面与$AC$交于点$G$, 求证:$G$是$AC$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,2,0) node [above] {$A$} coordinate (A);\n\\draw (2.2,-1.2,0) node [below] {$C$} coordinate (C);\n\\draw ($(A)!0.5!(D)$) node [right] {$F$} coordinate (F);\n\\draw ($(B)!0.5!(C)$) node [below] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(C)$) node [right] {$G$} coordinate (G);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (A) -- (C);\n\\draw [dashed] (B) -- (D);\n\\draw (E) -- (G) -- (F);\n\\draw [dashed] (E) -- (F);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332230,7 +333254,9 @@ "id": "020916", "content": "如图, 已知$A$、$B$、$C$、$D$是不共面的四个点, $M$、$N$分别是$\\triangle ACD$、$\\triangle BCD$的重心. 试判断平面$ABC$、平面$ACD$、平面$BCD$、平面$ABD$中, 哪些平面与$MN$平行, 并说明理由.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0,0) node [right] {$D$} coordinate (D);\n\\draw (2,2,0) node [above] {$A$} coordinate (A);\n\\draw (2.2,-1.2,0) node [below] {$C$} coordinate (C);\n\\filldraw (barycentric cs:A=1,C=1,D=1) node [above] {$M$} circle (0.03) coordinate (M);\n\\filldraw (barycentric cs:B=1,C=1,D=1) node [below] {$N$} circle (0.03) coordinate (N);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle (A) -- (C);\n\\draw [dashed] (B) -- (D) (M) -- (N);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332249,7 +333275,9 @@ "id": "020917", "content": "``直线$l$垂直于平面$\\alpha$内的无数条直线''是``$l \\perp \\alpha$''的\\bracket{20}.\n\\twoch{充分不必要条件}{必要不充分条件}{充要条件}{既不充分又不必要条件}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332268,7 +333296,9 @@ "id": "020918", "content": "在正方体$ABCD-A_1B_1C_1D_1$中找出表示下列距离的线段:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 点$A_1$到直线$BC$的距离为\\blank{50};\\\\\n(2) 点$A$到平面$B_1BCC_1$的距离为\\blank{50};\\\\\n(3)$B_1D_1$和平面$ABCD$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332287,7 +333317,9 @@ "id": "020919", "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw [dashed] (C) -- (D1) (B) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 点$A$到$CD_1$的距离为\\blank{50};\\\\\n(2) 点$A$到$BD_1$的距离为\\blank{50};\\\\\n(3) 点$A$到面$BB_1D_1D$的距离为\\blank{50};\\\\\n(4)$AA_1$和面$BB_1D_1D$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332306,7 +333338,9 @@ "id": "020920", "content": "有一根旗杆$AB$高$8$米, 它的顶端挂一条长$10$米的绳子, 拉紧绳子并把它的下端放在地面上的两点(和旗杆脚不在同一直线上)$C$、$D$. 如果这两个点和旗杆脚 $B$的距离都是$6$米, 那么旗杆就和地面垂直, 为什么?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332325,7 +333359,9 @@ "id": "020921", "content": "将一段长$12 \\text{cm}$的铁丝折成两两互相垂直的三段, 长度分别为$3 \\text{cm}$、$4 \\text{cm}$和$5 \\text{cm}$, 则原铁丝两个端点的距离为\\blank{50}$\\text{cm}$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332344,7 +333380,9 @@ "id": "020922", "content": "与空间不共面的四点等距离的平面共有\\bracket{20}.\n\\fourch{$3$个}{$4$个}{$6$个}{$7$个}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332363,7 +333401,9 @@ "id": "020923", "content": "如图所示, 在正方体$ABCD-A_1B_1C_1D_1$的侧面$AB_1$内有一动点$P$到直线$AB$与直线$B_1C_1$的距离相等, 则动点$P$所在曲线的形状为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) ++ (0,-\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,-\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,-\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,-\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D1) -- (D);\n\\draw (A1) -- (B1) -- (C1);\n\\draw [dashed] (A1) -- (D1) -- (C1);\n\\filldraw ($(B)!0.5!(B1)$) ++ (-0.8,-0.16) circle (0.03) node [right] {$P$} coordinate (P);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (2,0) node [below] {$B_1$} coordinate (B_1);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw (2,2) node [above] {$B$} coordinate (B);\n\\draw (A_1) rectangle (B);\n\\draw ($(B)!0.5!(B_1)$) coordinate (O);\n\\draw (A_1) -- (O);\n\\filldraw ($(A_1)!0.4!(O)$) circle (0.03) node [above] {$P$} coordinate (P);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (2,0) node [below] {$B_1$} coordinate (B_1);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw (2,2) node [above] {$B$} coordinate (B);\n\\draw (A_1) rectangle (B);\n\\draw (A_1) arc (180:0:1);\n\\filldraw ($(A_1)!0.5!(B_1)$) ++ (125:1) circle (0.03) node [above] {$P$} coordinate (P);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (2,0) node [below] {$B_1$} coordinate (B_1);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw (2,2) node [above] {$B$} coordinate (B);\n\\draw (A_1) rectangle (B);\n\\draw [domain = -2:0] plot ({\\x+2},{1-\\x*\\x/4});\n\\filldraw (0.6,0.51) circle (0.03) node [above] {$P$} coordinate (P);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [below] {$A_1$} coordinate (A_1);\n\\draw (2,0) node [below] {$B_1$} coordinate (B_1);\n\\draw (0,2) node [above] {$A$} coordinate (A);\n\\draw (2,2) node [above] {$B$} coordinate (B);\n\\draw (A_1) rectangle (B);\n\\draw ($(B)!0.5!(B_1)$) coordinate (O);\n\\draw (O) arc (90:180:1);\n\\filldraw (B_1) ++ (140:1) circle (0.03) node [above] {$P$} coordinate (P);\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332382,7 +333422,9 @@ "id": "020924", "content": "已知矩形$ABCD$所在平面外一点$P$, $PA \\perp$平面$ABCD$, $E$、$F$分别是$AB$、$PC$的中点. 求证: $EF \\perp CD$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332401,7 +333443,9 @@ "id": "020925", "content": "已知$SA \\perp$平面$ABC$, $\\angle ABC=90^{\\circ}$, $AM \\perp SB$, $AN \\perp SC$. 求证:$SC \\perp$平面$AMN$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$C$} coordinate (C);\n\\draw (0,2,0) node [above] {$S$} coordinate (S);\n\\draw (1.92,0,1.44) node [below] {$B$} coordinate (B);\n\\draw ($(S)!(A)!(C)$) node [above right] {$N$} coordinate (N);\n\\draw ($(S)!{25/61}!(B)$) node [left] {$M$} coordinate (M);\n\\draw (A) -- (M) -- (N) (S) -- (B);\n\\draw (S) -- (A) -- (B) -- (C) -- cycle;\n\\draw [dashed] (A) -- (C) (A) -- (N);\n\\draw (M) pic [draw, scale = 0.3] {right angle = A--M--B};\n\\draw (N) pic [draw, scale = 0.3] {right angle = A--N--C};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332420,7 +333464,9 @@ "id": "020926", "content": "判断下列命题的真假, 并说明理由:\\\\\n(1) 一条直线在平面内的投影是一条直线;\\\\\n(2) 在平面内投影是直线的图形一定是直线;\\\\\n(3) 如果两条线段在同一平面内的投影长相等, 那么这两条线段的长相等;\\\\\n(4) 如果两条斜线与平面所成的角相等, 那么这两条斜线互相平行.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332439,7 +333485,9 @@ "id": "020927", "content": "直线$l$与平面$\\beta$所成角为$\\theta$, 则$\\theta$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332458,7 +333506,9 @@ "id": "020928", "content": "正方体$ABCD-A_1B_1C_1D_1$中, 直线$AB_1$与平面$AA_1C_1C$所成角等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332477,7 +333527,9 @@ "id": "020929", "content": "等腰直角三角形$ABC$一直角边在平面$\\alpha$内, 斜边与平面$\\alpha$成$30^{\\circ}$, 则另一直角边与平面$\\alpha$所成角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332496,7 +333548,9 @@ "id": "020930", "content": "过平面$\\alpha$外一点$P$的斜线段$PA$的长是过这点的垂线段$PB$的$\\dfrac{2 \\sqrt 3}3$倍($A, B \\in \\alpha$), 求斜线$PA$与平面$\\alpha$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332515,7 +333569,9 @@ "id": "020931", "content": "如图, 在长方体$ABCD-A_1B_1C_1D_1$中, $AB=BC=4$, $A_1A=5, M$是$AB$的中点. 求直线$C_1M$与平面$ABCD$所成的角的大小. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\def\\l{4}\n\\def\\m{4}\n\\def\\n{5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(B)$) node [below] {$M$} coordinate (M);\n\\draw [dashed] (M) -- (C1);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332534,7 +333590,9 @@ "id": "020932", "content": "$PA$、$PB$是平面$\\alpha$的斜线, 已知$\\angle APB=90^{\\circ}$, $AB=10$, 点$P$到平面$PA$和平面$\\alpha$所成的角为$30^{\\circ}$, 求$PB$与平面$\\alpha$所成的角的大小.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5, z = {(-45:0.5cm)}]\n\\draw (0,0,0) coordinate (O);\n\\draw (0,3,0) node [above] {$P$} coordinate (P);\n\\draw ({-3*sqrt(3)},0,0) node [left] {$A$} coordinate (A);\n\\draw ({sqrt(3)},0,{2*sqrt(13)}) node [right] {$B$} coordinate (B);\n\\draw [name path = AP] (A) -- (P);\n\\draw [name path = BP] (B) -- (P);\n\\draw (A) -- (B) node [midway, below] {$10$};\n\\draw [dashed] (B) -- (P) (O) -- (B);\n\\draw [dashed] (P) -- (O) node [midway, left] {$3$} -- (A);\n\\draw (A) pic [draw, \"$30^\\circ$\", angle eccentricity = 1.8]{angle = O--A--P};\n\\draw (P) pic [draw, scale = 0.3] {right angle = A--P--B};\n\\draw (O) pic [draw, scale = 0.3] {right angle = A--O--P};\n\\draw (O) pic [draw, scale = 0.3] {right angle = B--O--P};\n\\draw (B) ++ (0.5,-0.5) ++ (4,4) coordinate (R) --++ (-4,-4) --++ (-14,0) --++ (4,4) coordinate (L);\n\\path [name path = LR] (L) -- (R);\n\\path [name intersections = {of = LR and AP, by = M}];\n\\path [name intersections = {of = LR and BP, by = N}];\n\\draw (L) -- (M) (R) -- (N);\n\\draw [dashed] (M) -- (N);\n\\draw (R) ++ (-1,0) node [below] {$\\alpha$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332553,7 +333611,9 @@ "id": "020933", "content": "正方体$ABCD-A_1B_1C_1D_1$中, $E$、$F$分别是边$AB$、$BC$的中点, $Q$为$EF$的中点, 设$B_1Q$与底面$ABCD$的夹角为$\\varphi$, 则$\\tan \\varphi$的值为\\bracket{20}.\n\\fourch{$\\dfrac{\\sqrt 2}4$}{$2 \\sqrt 2$}{$2$}{$\\dfrac 12$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332572,7 +333632,9 @@ "id": "020934", "content": "设直线$l$和平面$\\alpha$所成的角为$30^{\\circ}$, 则直线$l$与平面$\\alpha$内所有直线所成的角中最大的角等于\\bracket{20}.\n\\fourch{$150^{\\circ}$}{$30^{\\circ}$}{$90^{\\circ}$}{$60^{\\circ}$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332591,7 +333653,9 @@ "id": "020935", "content": "若斜线$l$和平面$\\gamma$所成的角为$\\alpha$, $l$与此平面内某直线所成的角为$\\beta$, 则\\bracket{20}.\n\\twoch{$\\alpha \\leq \\beta$}{$\\alpha=\\beta$}{$\\alpha \\geq \\beta$}{$\\alpha$与$\\beta$的大小关系不确定}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332610,7 +333674,9 @@ "id": "020936", "content": "两条异面直线在平面$\\alpha$内的投影不可能是\\bracket{20}.\n\\twoch{两条平行直线}{两条相交直线}{一条直线及直线外一点}{两个点}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332629,7 +333695,9 @@ "id": "020937", "content": "已知$P$是等边三角形$ABC$所在平面外一点, $PA=PB=PC=\\dfrac 23$, $\\triangle ABC$的边长为$1$, 求$PC$和平面$ABC$所成的角的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332648,7 +333716,9 @@ "id": "020938", "content": "矩形$ABCD$中, $AB=2$, $BC=1$, $PC \\perp$面$ABCD$, $PC=2$, 则点$P$到直线$BD$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332667,7 +333737,9 @@ "id": "020939", "content": "$P$是$\\triangle ABC$所在平面$\\alpha$外一点, 过$P$作$PO \\perp$平面$\\alpha$, 垂足是$O$, 联结$PA$, $PB$, $PC$.\\\\\n(1) 若$PA=PB=PC$, 则$O$为$\\triangle ABC$的\\blank{20}心;\\\\\n(2) 若$PA=PB=PC$, 且$\\angle ACB=90^{\\circ}$, 则$O$是$AB$边的\\blank{20}点;\\\\\n(3) 若$PA \\perp PB$, $PA \\perp PC$, $PC \\perp PB$, 则$O$是$\\triangle ABC$的\\blank{20}心;\\\\\n(4) 若$P$到三边$AB, BC, CA$的距离相等, 且$P$在$\\triangle ABC$内部, 则$O$是$\\triangle ABC$的\\blank{20}心.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332686,7 +333758,9 @@ "id": "020940", "content": "如图, $PA \\perp$平面$ABC$, $\\triangle ABC$中, $BC \\perp AC$, 则图中直角三角形的个数是\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (3,0,0) node [right] {$B$} coordinate (B);\n\\draw (0,2,0) node [above] {$P$} coordinate (P);\n\\draw (1.92,0,1.44) node [below] {$C$} coordinate (C);\n\\draw (S) -- (A) -- (C) -- (B) -- cycle (C) -- (P);\n\\draw [dashed] (A) -- (B);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$4$}{$3$}{$2$}{$1$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -332705,7 +333779,9 @@ "id": "020941", "content": "设三棱锥$P-ABC$的顶点$P$在平面$ABC$上的投影是$H$, 给出以下四个命题:\\\\\n\\textcircled{1} 若$PA \\perp BC, PB \\perp AC$, 则$H$是$\\triangle ABC$的垂心;\\\\\n\\textcircled{2} 若$PA, PB, PC$两两互相垂直, 则$H$是$\\triangle ABC$的垂心;\\\\\n\\textcircled{3} 若$\\angle ABC=90^{\\circ}, H$是$AC$的中点, 则$|PA|=|PB|=|PC|$;\\\\\n\\textcircled{4} 若$|PA|=|PB|=|PC|$, 则$H$是$\\triangle ABC$的外心. 其中正确命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332724,7 +333800,9 @@ "id": "020942", "content": "已知点$C$在$\\angle AOB$所确定的平面$\\alpha$外, 设点$C$在平面$\\alpha$内的投影为点$H$. 若$\\angle COA=\\angle COB$, 证明: 点$H$在$\\angle AOB$的角平分线上.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332743,7 +333821,9 @@ "id": "020943", "content": "已知点$P$是$\\triangle ABC$所在平面$\\alpha$外一点, 若$\\angle BAC=90^{\\circ}$, 且$\\angle PAB=\\angle PAC=60^{\\circ}$, 求直线$PA$与平面$\\alpha$的夹角.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332762,7 +333842,9 @@ "id": "020944", "content": "从平面$\\alpha$外一点$P$向$\\alpha$引垂线$PO$及斜线$PA$、$PB$, $O$是垂足, $A$、$B$是斜足. 设$PA$与$\\alpha$的夹角为$\\theta_1, PB$与$\\alpha$的夹角为$\\theta_2$.\\\\\n(1) 若$|PA|<|PB|$, 比较$\\theta_1$与$\\theta_2$的大小;\\\\\n(2) 若线段$PA$、$PB$在平面$\\alpha$内的投影分别长为$2$、$12$, 且$\\theta_1-\\theta_2=\\dfrac{\\pi}4$, 求$|PO|$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332781,7 +333863,9 @@ "id": "020945", "content": "下列命题中不正确的是\\blank{50}.\\\\\n\\textcircled{1} 若平面$\\alpha$与平面$\\beta$平行, 则平面$\\alpha$内任一直线平行于平面$\\beta$;\\\\\n\\textcircled{2} 如果一个平面内任何一条直线都平行于另一个平面, 那么这两个平面平行;\\\\\n\\textcircled{3} 如果一条直线$m$与两个平面$\\alpha$、$\\beta$所成的角相等, 那么$\\alpha\\parallel \\beta$;\\\\\n\\textcircled{4} 分别在两个平行平面内的两条直线只能是平行直线或异面直线.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332800,7 +333884,9 @@ "id": "020946", "content": "简答题(需说明理由):\\\\\n(1) 平行于同一直线的两个平面是否一定平行?\\\\\n(2) 两个平面分别经过两条平行直线, 这两个平面是否一定平行?\\\\\n(3) 分别在两个平行平面内的两条直线是否一定平行?", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332819,7 +333905,9 @@ "id": "020947", "content": "在正方体$ABCD-A'B'C'D'$中, $E$、$F$、$G$分别是$AD$、$DD'$、$DC$的中点, 求证: 平面$EFG$平行于平面$A'BC'$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A'$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B'$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C'$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D'$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1) -- (B) -- (C1) -- cycle;\n\\draw ($(A)!0.5!(D)$) node [left] {$E$} coordinate (E);\n\\draw ($(C)!0.5!(D)$) node [below] {$G$} coordinate (G);\n\\draw ($(D)!0.5!(D1)$) node [left] {$F$} coordinate (F);\n\\draw [dashed] (E) -- (F) -- (G) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332838,7 +333926,9 @@ "id": "020948", "content": "在正方体$ABCD-A_1B_1C_1D_1$中, $O$是$A_1B$的中点, 求证: 直线$DO\\parallel$平面$B_1D_1C$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332857,7 +333947,9 @@ "id": "020949", "content": "$\\alpha$、$\\beta$是两个不重合的平面, $a$、$b$是两条不同的直线, 在下列条件中可判定$\\alpha\\parallel \\beta$的是\\blank{50}.\\\\\n\\textcircled{1} 平面$\\alpha$、$\\beta$都平行于直线$a$、$b$;\\\\\n\\textcircled{2} 平面$\\alpha$内有三个不共线的点到平面$\\beta$的距离相等;\\\\\n\\textcircled{3} $a$、$b$是平面$\\alpha$内的两条直线, 且$a\\parallel \\beta$, $b\\parallel \\beta$;\\\\\n\\textcircled{4} $a$、$b$是两条异面直线, 且$a\\parallel \\alpha$, $b\\parallel \\alpha$, $a\\parallel \\beta$, $b\\parallel \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332876,7 +333968,9 @@ "id": "020950", "content": "设$a$、$b$是异面直线, 直线$a$在平面$\\alpha$内, 直线$b$在平面$\\beta$内, 且$a\\parallel \\beta$, $b\\parallel \\alpha$, 求证: $\\alpha\\parallel \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332895,7 +333989,9 @@ "id": "020951", "content": "已知不共面的三条直线$a$、$b$、$c$相交于点$P$, 平面$\\alpha$、$\\beta$与直线$a$、$b$、$c$分别相交于$A$、$B$、$C$和$A_1$、$B_1$、$C_1$, $P$是线段$AA_1,BB_1,CC_1$的一个交点, 且$\\dfrac{PA}{PA_1}=\\dfrac{PB}{PB_1}=\\dfrac{PC}{PC_1}$, 求证: $\\alpha\\parallel \\beta$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -332914,7 +334010,9 @@ "id": "020952", "content": "一山坡的倾角为$30^{\\circ}$, 如果在山坡上沿着一条与斜坡坡脚成$45^{\\circ}$角的直路前进$1000$米, 则升高了\\blank{50}米.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332933,7 +334031,9 @@ "id": "020953", "content": "二面角$\\alpha-l-\\beta$的面$\\alpha$内有一条直线$AB$, 它与$l$的夹角为$\\dfrac{\\pi}4$, 与平面$\\beta$所成角为$\\dfrac{\\pi}6$, 则二面角的大小是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332952,7 +334052,9 @@ "id": "020954", "content": "设$S$为$\\triangle ABC$所在平面外一点, 平面$SAB$、$SBC$、$SCA$与$\\triangle ABC$所在平面所成的二面角都相等, 则点$S$在平面$ABC$内的射影是$\\triangle ABC$的\\blank{20}心.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332971,7 +334073,9 @@ "id": "020955", "content": "已知$E$、$F$分别是正方体$ABCD-A_1B_1C_1D_1$的棱$BC, CC_1$的中点, 则截面$AEFD_1$与底面$ABCD$所成二面角的正弦值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -332990,7 +334094,9 @@ "id": "020956", "content": "已知二面角$\\alpha-AB-\\beta$为$30^{\\circ}$, $P$是面$\\alpha$内一点, 点$P$到面$\\beta$的距离是$1$, 求点$P$在面$\\beta$内的投影到$AB$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333009,7 +334115,9 @@ "id": "020957", "content": "在正方体中$ABCD-A_1B_1C_1D_1$中, $E_1$为$A_1D_1$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-150:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A1)!0.5!(D1)$) node [above left] {$E_1$} coordinate (E1);\n\\draw [dashed] (A) -- (E1) (B) -- (E1) (A) -- (D1);\n\\draw (A) -- (B1) (B1) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 求二面角$E_1-AB-C$的大小;\\\\\n(2) 求二面角$C_1-B_1D_1-A$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333028,7 +334136,9 @@ "id": "020958", "content": "如图, 长方体$ABCD-A_1B_1C_1D_1$中, $AB=\\sqrt 3$, $AA_1=1$, 截面$AB_1C_1D$为正方形. 求:\n\\begin{center}\n\\begin{tikzpicture}[>=latex, z = {(-150:0.5cm)}]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [above] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B) -- (C1) (A) -- (B1);\n\\draw [dashed] (A) -- (C1) (A) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 点$B_1$到平面$ABC_1$的距离;\\\\\n(2) 二面角$C_1-AB-B_1$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333047,7 +334157,9 @@ "id": "020959", "content": "已知$P$是二面角$\\alpha-AB-\\beta$内一点, $PC \\perp \\alpha$, 垂足为$C$, $PD \\perp \\beta$, 垂足为$D$, 且$PC=3$, $PD=4$, $\\angle CPD=60^{\\circ}$.\\\\\n(1) 求二面角$\\alpha-AB-\\beta$的大小;\\\\\n(2) 求$CD$的长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333066,7 +334178,9 @@ "id": "020960", "content": "在四面体$ABCD$中, $BD \\perp$平面$ACD$, 且$|AD|=|DC|=|CA|=|DB|=1$.\\\\\n(1) 求证: 平面$ACD \\perp$平面$BCD$;\\\\\n(2) 求二面角$A-BC-D$的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333085,7 +334199,9 @@ "id": "020961", "content": "如图, 已知空间四边形$ABCD$中, $BC=AC$, $AD=BD$, $E$是$AB$的中点.\n求证: 平面$CDE\\perp$平面$ABC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [left] {$B$} coordinate (B);\n\\draw (4,0,0) node [right] {$C$} coordinate (C);\n\\draw ({21/8},0,{3*sqrt(15)/8}) node [below] {$D$} coordinate (D);\n\\draw ({(289-3*sqrt(1065))/64},2,{(-9*sqrt(15)-11*sqrt(71))/64}) node [above] {$A$} coordinate (A);\n\\draw (B) -- (D) -- (C) -- (A) -- cycle;\n\\draw ($(A)!0.5!(B)$) node [above left] {$E$} coordinate (E);\n\\draw (D) -- (E) (D) -- (A);\n\\draw [dashed] (E) -- (C) (B) -- (C);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333104,7 +334220,9 @@ "id": "020962", "content": "如图, 在正方体$ABCD-A'B'C'D'$中. 求证:\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A'$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B'$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C'$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D'$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1) -- (C1) -- (B) -- cycle;\n\\end{tikzpicture}\n\\end{center}\n(1) 平面$B' DB \\perp$平面$A'C'B$;\\\\\n(2) 求二面角$B-A' C'-B'$的正弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333123,7 +334241,9 @@ "id": "020963", "content": "如图, 已知四棱锥$P-ABCD$的底面是直角梯形, $\\angle ABC=\\angle BCD=90^{\\circ}$, $AB=BC=PB=PC=2CD$, 侧面$PBC \\perp$底面$ABCD$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0,0) node [below] {$A$} coordinate (A);\n\\draw (2,0,0) node [below] {$B$} coordinate (B);\n\\draw (2,0,-2) node [right] {$C$} coordinate (C);\n\\draw (1,0,-2) node [below] {$D$} coordinate (D);\n\\draw ($(B)!0.5!(C)$) ++ (0,{sqrt(3)},0) node [above] {$P$} coordinate (P);\n\\draw (A) -- (B) -- (C) (P) -- (A) (P) -- (B) (P) -- (C);\n\\draw [dashed] (A) -- (D) -- (C) (D) -- (P);\n\\end{tikzpicture}\n\\end{center}\n(1) $PA$与$BD$是否相互垂直, 请证明你的结论;\\\\\n(2) 求二面角$P-BD-C$的大小;\\\\\n(3) 求证: 平面$PAD \\perp$平面$PAB$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333142,7 +334262,9 @@ "id": "020964", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, 找出下列异面直线的公垂线段.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\def\\m{1.5}\n\\def\\n{1.5}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\m) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\m) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\n,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\n,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\n,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\n,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (B) -- (C1);\n\\end{tikzpicture}\n\\end{center}\n(1) $AB$和$DD_1$;\\\\\n(2) $AA_1$和$BC_1$.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333161,7 +334283,9 @@ "id": "020965", "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$1$, 则异面直线$A_1D$与$AC$之间的距离是\\blank{50}; 若$BD_1=2 \\sqrt 3$, 则异面直线$BD_1$与$AC$之间的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333180,7 +334304,9 @@ "id": "020966", "content": "已知长方体$ABCD-A_1B_1C_1D_1$中, 棱$AA_1=5$, $AB=12$, $BC=5$, 那么直线$B_1C_1$到平面$A_1BCD_1$的距离是\\blank{50}, 异面直线$AB$与$B_1C_1$距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333199,7 +334325,9 @@ "id": "020967", "content": "在长方体$ABCD-A_1B_1C_1D_1$中, $|AA_1|=3$, $|AB|=4$, $|BC|=5$. 则\\\\\n(1) $A$、$C_1$两点间的距离为\\blank{50};\\\\\n(2) 点$A$到直线$B_1C_1$的距离为\\blank{50};\\\\\n(3) 直线$AB$到平面$A_1B_1C_1D_1$的距离为\\blank{50};\\\\\n(4) 异面直线$AD$与$A_1B_1$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333218,7 +334346,9 @@ "id": "020968", "content": "已知一个四面体的$6$条棱长分别为$1$、$1$、$1$、$1$、$\\sqrt 2$、$a$, 且长为$a$的棱与长为$\\sqrt 2$的棱异面, 则$a$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333237,7 +334367,9 @@ "id": "020969", "content": "在空间四边形$ABCD$中, $AB=DC=4$, $BC=AD=3$, $AD \\perp BC$, $AB \\perp BC$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.7]\n\\draw (0,0,0) node [left] {$A$} coordinate (A);\n\\draw (5,0,0) node [right] {$C$} coordinate (C);\n\\draw ({16/5},0,{12/5}) node [below] {$B$} coordinate (B);\n\\draw ({9/5},{3*sqrt(7)/4},{27/20}) node [above] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (B) -- (D);\n\\draw [dashed] (A) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 求证:$BD$是$AD$、$BC$的公垂线;\\\\\n(2) 求$BD$的长.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333256,7 +334388,9 @@ "id": "020970", "content": "长方体$ABCD-A_1B_1C_1D_1$中, 已知$AB=a, BC=b, AA_1=c$, 且$a>b$, 求:\\\\\n(1) 下列异面直线之间的距离: $AB$与$CC_1$, $AB$与$A_1C_1$, $AB$与$B_1C$;\\\\\n(2) 异面直线$D_1B$与$AC$所成角的余弦值.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333275,7 +334409,9 @@ "id": "020971", "content": "判断下列说法是否正确. 如果正确, 请说明理由; 如果不正确, 请举一个反例.\\\\\n(1) 有两个相邻的侧面是矩形的棱柱是直棱柱;\\\\\n(2) 正四棱柱是正方体;\\\\\n(3) 直四棱柱是长方体.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333294,7 +334430,9 @@ "id": "020972", "content": "长方体$ABCD-A_1B_1C_1D_1$中同一顶点三条棱长分别为$a$、$b$、$c$, 则它的体对角线$AC_1$的长为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333313,7 +334451,9 @@ "id": "020973", "content": "集合$M$是所有正四棱柱组成的集合, 集合$N$是所有长方体组成的集合, 集合$Q$是所有正方体组成的集合, 集合$P$是所有直四棱柱组成的集合, 则它们的包含关系为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333332,7 +334472,9 @@ "id": "020974", "content": "现有以下三个命题: \\textcircled{1} 底面是平行四边形的四棱柱是平行六面体; \\textcircled{2} 底面是矩形的平行六面体是长方体; \\textcircled{3} 直四棱柱是直平行六面体. 其中真命题的序号是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333351,7 +334493,9 @@ "id": "020975", "content": "下列命题中是假命题的是\\bracket{20}.\n\\twoch{直棱柱的侧棱是棱柱的高}{有一侧面是矩形的棱柱是直棱柱}{直棱柱的侧面为矩形}{有一条侧棱垂直于底面的棱柱是直棱柱}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -333370,7 +334514,9 @@ "id": "020976", "content": "将下列表格按要求填写:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline & 斜棱柱 & 直棱柱 & 正三棱柱 & 四棱柱 \\\\\n\\hline 底面形状 & \\blank{50} & \\blank{50}& \\blank{50}& \\blank{50}\\\\\n\\hline 侧棱与底面的关系 &\\blank{50} &\\blank{50} & \\blank{50}& \\blank{50}\\\\\n\\hline 侧面形状 &\\blank{50} & \\blank{50}&\\blank{50} & \\blank{50}\\\\\n\\hline 对角面形状 & \\blank{50}& \\blank{50}& \\blank{50}&\\blank{50} \\\\\n\\hline 平行底面的截面与底面的关系 & \\blank{50}&\\blank{50} &\\blank{50} &\\blank{50} \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333389,7 +334535,9 @@ "id": "020977", "content": "如图都是正方体的表面展开图, 还原成正方体后, 其中两个完全一样的是 \\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw (0,0) node [below] {\\textcircled{1}};\n\\draw (-0.5,1) --++ (1,0) --++ (0,3) --++ (-2,0) --++ (0,-1) --++ (4,0) --++ (0,-1) --++ (-3,0) ++ (0,-1) --++ (0,3) ++ (2,-1) --++ (0,-1);\n\\draw (0,1.5) node {d} ++ (0,1) node {c} ++ (0,1) node {b} ++ (-1,0) node {a} ++ (2,-1) node {e} ++ (1,0) node {f};\n\\draw (5,0) node [below] {\\textcircled{2}};\n\\draw (4.5,0) --++ (1,0) --++ (0,4) --++ (1,0) --++ (0,-1) --++ (-3,0) --++ (0,1) --++ (2,0) ++ (-1,0) --++ (0,-4) ++ (0,1) --++ (1,0) ++ (-1,1) --++ (1,0);\n\\draw (5,0.5) node {b} ++ (0,1) node {c} ++ (0,1) node {e} ++ (0,1) node {f} ++ (-1,0) node {d} ++ (2,0) node {a};\n\\draw (10,0) node [below] {\\textcircled{3}};\n\\draw (8.5,0) --++ (2,0) --++ (0,4) --++ (-1,0) --++ (0,-4) ++ (-1,0) --++ (0,1) --++ (2,0) ++ (-1,1) --++ (2,0) --++ (0,1) --++ (-2,0);\n\\draw (10,0.5) node {b} ++ (-1,0) node {c} ++ (1,1) node {d} ++ (0,1) node {e} ++ (0,1) node {a} ++ (1,-1) node {f};\n\\draw (15,0) node [below] {\\textcircled{4}};\n\\draw (14.5,0) --++ (-1,0) --++ (0,2) --++ (3,0) --++ (0,1) --++ (-2,0) --++ (0,1) --++ (1,0) --++ (0,-3) --++ (-2,0) ++ (1,-1) --++ (0,3);\n\\draw (14,0.5) node {e} ++ (0,1) node {c} ++ (1,0) node {a} ++ (0,1) node {b} ++ (1,0) node {d} ++ (-1,1) node {f};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{\\textcircled{1}\\textcircled{2}}{\\textcircled{2}\\textcircled{3}}{\\textcircled{3}\\textcircled{4}}{\\textcircled{1}\\textcircled{4}}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -333408,7 +334556,9 @@ "id": "020978", "content": "平面上矩形$ABCD$的对角线$AC$与边$AB$和$AD$所成角分别为$\\alpha$、$\\beta$, 则$\\cos ^2 \\alpha+\\cos ^2 \\beta=1$. 类比空间可得怎样类似的命题? 证明你的命题.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333427,7 +334577,9 @@ "id": "020979", "content": "柱体(棱柱、圆柱)体积公式$V=$\\blank{50}, 等底等高的柱体体积\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333446,7 +334598,9 @@ "id": "020980", "content": "一个正四棱柱的侧面展开图是一个边长为$4$的正方形, 则它的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333465,7 +334619,9 @@ "id": "020981", "content": "一个长方体的长、宽、高的比为$1: 2: 3$, 对角线长是$2 \\sqrt {14}$, 则它的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333484,7 +334640,9 @@ "id": "020982", "content": "已知三棱柱的底面是$\\triangle ABC$, $AB=13 \\text{cm}$, $BC=5 \\text{cm}$, $CA=12 \\text{cm}$, 侧棱$AA'$的长是$20 \\text{cm}$, 且侧棱$AA'$与底面所成的角为$60^{\\circ}$, 求这个三棱柱的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333503,7 +334661,9 @@ "id": "020983", "content": "在万吨水压机上, 有四根圆柱形钢柱, 高$18$米, 内径$0.4$米, 外径$1$米, 求这四根钢柱的总质量. (结果精确到$1$吨, 钢的密度为$7.9$克/立方厘米)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333522,7 +334682,9 @@ "id": "020984", "content": "有一堆相同规格的六角螺帽毛坯共重$5.8$千克, 已知每个六角螺帽毛坏的底面六边形的边长是$12$毫米, 高$10$毫米, 内孔直径是$10$毫米, 共有毛坯多少个? (铁的密度为$7.8$克/立方厘米)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333541,7 +334703,9 @@ "id": "020985", "content": "已知正六棱柱最长的一条对角线长为$13$厘米, 侧面积为$180$平方厘米, 求这个棱柱的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333560,7 +334724,9 @@ "id": "020986", "content": "已知电镀螺杆的尺寸如图所示(图中单位: 毫米). 如果每平方米用锌$0.11$千克, 那么要电镀$100$个这样的螺杆需要多少锌?\n\\begin{center}\n\\begin{tikzpicture}[>=latex, scale = 0.1]\n\\draw (-6,0) -- (6,0) -- (6,5) -- (-6,5) -- cycle;\n\\draw (6,5) --++ (8,5) --++ (0,-5) -- (6,0);\n\\draw (-6,5) --++ (-8,5) coordinate (T) --++ (0,-5) -- (-6,0);\n\\draw (T) --++ (8,5) --++ (12,0) --++ (8,-5);\n\\draw (T) ++ (14,0) coordinate (O);\n\\filldraw [white] (O) ++ (5,0) rectangle++ (-10,25);\n\\draw (O) ++ (5,0) --++ (0,25);\n\\draw (O) ++ (-5,0) --++ (0,25);\n\\draw (O) ++ (5,0) arc (0:-180:5 and 2.5);\n\\draw (O) ++ (0,25) ellipse (5 and 2.5);\n\\draw (-6,-1) -- (-6,-3) (6,-1) -- (6,-3);\n\\draw [->] (-2,-2) -- (-6,-2);\n\\draw [->] (2,-2) -- (6,-2);\n\\draw (0,-2) node {$12$};\n\\draw (15,5) -- (19,5) (15,10) -- (19,10);\n\\draw (17,7.5) node {\\rotatebox{90}{$5$}};\n\\draw [->] (17,0) -- (17,5);\n\\draw [->] (17,20) -- (17,10);\n\\draw [->] (17,25) -- (17,35);\n\\draw (6,35) -- (19,35);\n\\draw (17,22.5) node {\\rotatebox{90}{$25$}};\n\\draw (-5,38) -- (-5,42) (5,38) -- (5,42);\n\\draw [->] (-10,40) -- (-5,40);\n\\draw [->] (10,40) -- (5,40);\n\\draw (0,40) node {$10$}; \n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333579,7 +334745,9 @@ "id": "020987", "content": "长方体长、宽、高的和为$14 \\text{cm}$, 对角线为$8 \\text{cm}$, 则其表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333598,7 +334766,9 @@ "id": "020988", "content": "圆柱侧面积为$S$, 底面周长为$C$, 则其体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333617,7 +334787,9 @@ "id": "020989", "content": "一圆柱高$h$, 侧面展开图中母线与对角线夹角为$60^{\\circ}$, 则其侧面积为\\blank{50}, 表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333636,7 +334808,9 @@ "id": "020990", "content": "已知直四棱柱的底面是边长分别为$5 \\text{cm}$、$6 \\text{cm}$. 且有一条对角线长为$8 \\text{cm}$的平行四边形, 该四棱柱最长的对角线为$10 \\text{cm}$, 求该四棱柱的侧面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333655,7 +334829,9 @@ "id": "020991", "content": "已知侧面积为$27$的正三棱柱的侧棱恰好是某个圆柱的三条母线, 且这个圆柱的底面半径为$2$, 求这个圆柱的表面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333674,7 +334850,9 @@ "id": "020992", "content": "已知长方体的长为$\\sqrt {29}$, 长、宽、高之和为$9$, 求这个长方体的表面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333693,7 +334871,9 @@ "id": "020993", "content": "已知长方体$ABCD-A_1B_1C_1D_1$的高为$h$, 底面积为$P$, 对角面$BB_1D_1D$的面积为$Q$, 求它的侧面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333712,7 +334892,9 @@ "id": "020994", "content": "已知正方体$ABCD-A'B'C'D'$的边长为$a$, 点$E$、$F$分别是棱$AD$、$AB$的中点.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A'$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B'$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C'$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D'$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw ($(A)!0.5!(D)$) node [left] {$E$} coordinate (E);\n\\draw ($(A)!0.5!(B)$) node [below] {$F$} coordinate (F);\n\\draw [dashed] (E) -- (F) (E) -- (D1);\n\\draw (F) -- (B1) (B1) -- (D1); \n\\end{tikzpicture}\n\\end{center}\n(1) 求证: 四边形$EFB' D'$是等腰梯形;\\\\\n(2) 求等腰梯形$EFB' D'$的面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333731,7 +334913,9 @@ "id": "020995", "content": "一个正方体和一个圆柱体等高, 且侧面积相等, 试比较它们的体积的大小.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333750,7 +334934,9 @@ "id": "020996", "content": "判断下列说法是否正确. 如果正确, 请说明理由; 如果不正确, 请举一个反例.\\\\\n(1) 底面是正多边形的棱锥是正棱锥;\\\\\n(2) 侧棱长柤等, 且底面是正多边形的棱锥是正棱锥;\\\\\n(3) 各侧面都是正三角形的四棱锥是正四棱锥;\\\\\n(4) ``三条侧棱两两互相垂直, 且侧棱与底面所成角都相等''是``棱锥为正三棱锥'' 的充要条件.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333769,7 +334955,9 @@ "id": "020997", "content": "若一个三棱锥的底面是直角三角形, 则这个三棱锥的三个侧面\\bracket{20}.\n\\twoch{都不是直角三角形}{至多只能有一个是直角三角形}{至多只能有两个是直角三角形}{可能都是直角三角形}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -333788,7 +334976,9 @@ "id": "020998", "content": "有下面五个命题:\n\\textcircled{1} 侧面都是全等的等腰三角形的棱锥是正棱锥; \\textcircled{2} 侧棱都相等的棱锥是正棱锥; \\textcircled{3} 底面是正方形的棱锥是正四棱锥; \\textcircled{4} 正四面体就是正四棱锥; \\textcircled{5} 顶点在底面上的射影既是底面多边形的内心, 又是底面多边形的外心的棱锥必是正棱锥.\n其中正确的命题是\\blank{50}.(写出所有正确的编号)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333807,7 +334997,9 @@ "id": "020999", "content": "已知棱锥的侧棱与底面所成的角都相等, 试说出棱锥的顶点在底面内的射影所在的位置, 并证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333826,7 +335018,9 @@ "id": "021000", "content": "已知棱锥的顶点在底面内的射影在底面的内部, 其侧面与底面所成的角都相等, 试说出棱锥的顶点在底面内的射影所在的位置, 并证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333845,7 +335039,9 @@ "id": "021001", "content": "已知正方体$ABCD-A_1B_1C_1D_1$的棱长为$2$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\end{tikzpicture}\n\\end{center}\n(1) 平面$DCB_1A_1$将正方体分割成两个多面体, 作出这两个多面体, 并说出它们的名称;\\\\\n(2) 平面$AB_1C_1$将直三棱柱$ABC-A_1B_1C_1$分制成两个多面体, 作出这两个多面体, 并说出它们的名称.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333864,7 +335060,9 @@ "id": "021002", "content": "如图, 正方体$ABCD-A_1B_1C_1D_1$中, 以$A_1,C_1,B_1,B$为顶点的四面体体积与正方体体积的比是\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\def\\l{2}\n\\draw (0,0,0) node [below left] {$A$} coordinate (A);\n\\draw (A) ++ (\\l,0,0) node [below right] {$B$} coordinate (B);\n\\draw (A) ++ (\\l,0,-\\l) node [right] {$C$} coordinate (C);\n\\draw (A) ++ (0,0,-\\l) node [left] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C);\n\\draw [dashed] (A) -- (D) -- (C);\n\\draw (A) ++ (0,\\l,0) node [left] {$A_1$} coordinate (A1);\n\\draw (B) ++ (0,\\l,0) node [right] {$B_1$} coordinate (B1);\n\\draw (C) ++ (0,\\l,0) node [above right] {$C_1$} coordinate (C1);\n\\draw (D) ++ (0,\\l,0) node [above left] {$D_1$} coordinate (D1);\n\\draw (A1) -- (B1) -- (C1) -- (D1) -- cycle;\n\\draw (A) -- (A1) (B) -- (B1) (C) -- (C1);\n\\draw [dashed] (D) -- (D1);\n\\draw (A1) -- (B) -- (C1) -- cycle;\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333883,7 +335081,9 @@ "id": "021003", "content": "正四棱锥底面积为$P$, 侧面积为$Q$, 体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333902,7 +335102,9 @@ "id": "021004", "content": "正六棱锥底面边长为$a$, 体积为$\\dfrac{\\sqrt 3}2 a^3$, 则侧棱与底面所成的角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -333921,7 +335123,9 @@ "id": "021005", "content": "如图所示, 过四面体一边及对边中点的截面, 把锥体分成两部分的体积比为\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (0,0) node [left] {$B$} coordinate (B);\n\\draw (3,0) node [right] {$D$} coordinate (D);\n\\draw (2,-1) node [below] {$C$} coordinate (C);\n\\draw (1.8,1.4) node [above] {$A$} coordinate (A);\n\\draw ($(C)!0.5!(D)$) node [right] {$E$} coordinate (E);\n\\draw (A) -- (B) -- (C) -- (D) -- cycle;\n\\draw (A) -- (C) (A) -- (E);\n\\draw [dashed] (B) -- (D) (B) -- (E);\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$2: 1$}{$1: 1$}{$3: 1$}{$2: 3$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -333940,7 +335144,9 @@ "id": "021006", "content": "直三棱柱$ABC-A_1B_1C_1$的体积为$V, P$、$Q$分别是$AA_1$、$CC_1$上的点, 且$AP=C_1Q$, 则四棱锥$B-APQC$的体积为\\bracket{20}.\n\\fourch{$\\dfrac 12V$}{$\\dfrac 13V$}{$\\dfrac 14V$}{$\\dfrac 23V$}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -333959,7 +335165,9 @@ "id": "021007", "content": "已知正三棱锥的侧棱长为$10$厘米, 侧棱与底面所成的角等于$\\arcsin \\dfrac 35$, 求这个三棱锥的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333978,7 +335186,9 @@ "id": "021008", "content": "有一个铜制工件, 它的下部分呈正四棱柱形, 上部分呈正四棱锥形, 且这个正四棱锥以正四棱柱的上底为底, 己知正四棱柱的底面边长是$50$毫米, 高是$10$毫米, 正四棱锥的侧面呈正三角形, 这个工件的质量是多少千克? (结果精确到$0.1$千克, 铜的密度是$8.9$克/立方厘米)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -333997,7 +335207,9 @@ "id": "021009", "content": "已知一个直角三角形的两条直角边分别为$15$厘米、$20$厘米, 以它的斜边为旋转轴旋转一周, 得旋转体, 求旋转体的体积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334016,7 +335228,9 @@ "id": "021010", "content": "一圆锥的高为$h$, 侧面展开图是半圆, 圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334035,7 +335249,9 @@ "id": "021011", "content": "一个圆柱的侧面展开图是正方形, 那么它的侧面积是底面积的\\blank{50}倍.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334054,7 +335270,9 @@ "id": "021012", "content": "将圆锥的母线长扩大到原来的$n$倍, 底面半径缩小为原来的$\\dfrac 1n$, 那么它的侧面积变为原来的\\blank{50}倍.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334073,7 +335291,9 @@ "id": "021013", "content": "已知正三棱锥的底面边长是$2$, 高是$4$, 则此正三棱锥的表面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334092,7 +335312,9 @@ "id": "021014", "content": "已知一个圆锥的高是$10 \\text{cm}$, 侧面展开图是半圆, 则这个圆锥的侧面积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334111,7 +335333,9 @@ "id": "021015", "content": "已知正四棱锥底面正方形的边长为$4 \\text{cm}$, 高与斜高的夹角为$30^{\\circ}$, 求正四棱锥的表面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334130,7 +335354,9 @@ "id": "021016", "content": "已知正三棱锥底面边长为$a$, 侧面与底面所成二面角是$45^{\\circ}$, 求它的表面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334149,7 +335375,9 @@ "id": "021017", "content": "在斜三棱柱$ABC-A_1B_1C_1$中, 底面是边长为$a$的正三角形, 侧棱长为$b, AA_1$与$AB$, $AC$都成$\\theta$角, 求这三棱柱的侧面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334168,7 +335396,9 @@ "id": "021018", "content": "圆锥$S$的底面半径$r=20 \\text{cm}, S$为圆锥的顶点, $O$为底面圆心, 如果底面半径$OQ$与母线$SA$垂直, $P$是$SA$的中点, 直线$PQ$与高$SO$所成的角为$\\alpha$, 且$\\tan \\alpha=2$. 求圆锥的表面积.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334187,7 +335417,9 @@ "id": "021019", "content": "一个直角梯形的上底、下底、高的比为$1: 2: \\sqrt 3$, 求由它旋转而成的圆台的上底面积, 下底面积和侧面积的比.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334206,7 +335438,9 @@ "id": "021020", "content": "过球面上两点的大圆的个数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334225,7 +335459,9 @@ "id": "021021", "content": "球$O$的小圆$O_1$的面积是$9 \\pi \\text{cm}$, 球的半径为$5 \\text{cm}$, 则球心$O$到该小圆$O_1$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334244,7 +335480,9 @@ "id": "021022", "content": "球的半径为$12 \\text{cm}$, 球的一个截面与球心的距离为$4 \\text{cm}$, 则截面圆的半径为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334263,7 +335501,9 @@ "id": "021023", "content": "如果球的大圆面积增为原来的$100$倍, 那么球的半径会变成原来的\\blank{50}倍.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334282,7 +335522,9 @@ "id": "021024", "content": "圆柱形容器内部盛有高度为$8 \\text{cm}$的水, 若放入三个相同的球 (球的半径与圆柱的底面半径相同) 后, 水恰好淹没最上面的球 (如右图所示), 则球的半径是\\blank{50}$\\text{cm}$.\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (0,3.5) -- (0,0) -- (1,0) -- (1,3.5);\n\\filldraw [fill = gray!50] (-0.4,0) rectangle (1.4,-0.5);\n\\filldraw [fill = gray!20] (0.5,0.5) circle (0.5);\n\\filldraw [fill = gray!20] (0.5,1.5) circle (0.5);\n\\filldraw [fill = gray!20] (0.5,2.5) circle (0.5);\n\\draw (0,3) -- (1,3);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334301,7 +335543,9 @@ "id": "021025", "content": "已知三个球的表面积之比是$1: 2: 3$, 这三个球的体积之比为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334320,7 +335564,9 @@ "id": "021026", "content": "体积为$V$的正方体的外接球的体积为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334339,7 +335585,9 @@ "id": "021027", "content": "球面上三点$A$、$B$、$C$, 且$AB=10$, $BC=24$, $AC=26$, 球的半径为 50 , 求球心到平面$ABC$的距离.", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334358,7 +335606,9 @@ "id": "021028", "content": "从一个底面半径和高都是$R$的圆柱中, 挖去一个以圆柱的上底为底、下底面的中心为顶点的圆锥, 得到一个如图所示的几何体. 如果用一个与圆柱下底面距离等于$d$并且平行于底面的平面去截这个几何体. 求截面面积.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (-2,0) arc (180:360:2 and 0.5);\n\\draw (-2,0) --++ (0,2) (2,0) --++ (0,2);\n\\draw (0,2) ellipse (2 and 0.5);\n\\draw [dashed] (-2,0) arc (180:0:2 and 0.5);\n\\draw [dashed] (-2,2) -- (0,0) -- (2,2);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334377,7 +335627,9 @@ "id": "021029", "content": "如图, 已知$OA$是球$O$的半径, $OA=5, O_2$是$OA$上的两点, 平面$\\alpha$、$\\beta$分别通过点$O_2$, 且垂直于$OA$, 截得圆$O_1$和圆$O_2$, 当圆$O_1$、圆$O_2$的面积分别为$9 \\pi$、$21 \\pi$时, 求$O_1O_2$的长.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.4]\n\\filldraw (0,0) circle (0.05) node [below] {$O$};\n\\filldraw (0,5) circle (0.05) node [above] {$A$};\n\\filldraw (0,2) circle (0.05) node [right] {$O_2$};\n\\filldraw (0,4) circle (0.05) node [right] {$O_1$};\n\\draw (0,0) circle (5);\n\\draw [dashed] ({-sqrt(21)},2) arc (180:0:{sqrt(21)} and {sqrt(21)/4});\n\\draw ({-sqrt(21)},2) arc (180:360:{sqrt(21)} and {sqrt(21)/4});\n\\draw [dashed] (-3,4) arc (180:0:3 and {3/4});\n\\draw (-3,4) arc (180:360:3 and {3/4});\n\\draw [dashed] (0,0) -- (0,5);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334396,7 +335648,9 @@ "id": "021030", "content": "有一个空心钢球, 重$142$克, 测得外径等于$5$厘米, 求它的内径. (结果精确到$0.1$厘米, 钢的密度为$7.9$克/立方厘米)", "objs": [], - "tags": [], + "tags": [ + "第六单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334415,7 +335669,10 @@ "id": "021031", "content": "一种旅行包上的号码锁有三个拨号盘, 每个拨号盘上有从$0$到$9$的$10$个数字, 这三个拨号盘可组成多少种不同的三位数号码?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334434,7 +335691,10 @@ "id": "021032", "content": "一个商场共有$9$个出入口, 若某人在进出商场时不要走同一个出入口, 则他一次进出商场共有多少种不同的进出法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334453,7 +335713,10 @@ "id": "021033", "content": "为了提高产品质量控制生产过程的温度、材料处理的时间和添加剂的剂量, 为此工厂进行生产试验. 试验控制温度有$150^{\\circ} \\text{C}$、$160^{\\circ} \\text{C}$和$170^{\\circ} \\text{C}$三种, 材料处理的时间有$10$分钟、$12$分钟两种, 添加剂的剂量有$2$克, $4$克和$6$克三种, 共需要做多少次试验?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334472,7 +335735,10 @@ "id": "021034", "content": "$(a_1+a_2+a_3+a_4)(b_1+b_2+b_3+b_4+b_5)(\\mathrm{C}_1+\\mathrm{C}_2)$展开后共有多少项?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334491,7 +335757,10 @@ "id": "021035", "content": "某个学校食堂准备了$5$种素菜、$3$种荤菜和$3$种汤, 取一种素菜、一种荤菜、一种汤配成一套菜, 这个学校食堂可以有多少套不同的菜?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334510,7 +335779,10 @@ "id": "021036", "content": "用$1$、$2$、$3$、$4$、$5$这五个数字可以组成多少个无重复数字的三位数的奇数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334529,7 +335801,10 @@ "id": "021037", "content": "要把$4$封信投入$3$个信箱, 共有多少种不同的投法? (允许将信全部或部分投入某一个信箱)", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334548,7 +335823,10 @@ "id": "021038", "content": "用$0$、$1$、$2$、$3$、$4$、$5$这六个数字可以组成多少个三位数? 多少个数字不重复的三位数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334567,7 +335845,10 @@ "id": "021039", "content": "整数$4620$的正约数有多少个? $4620$的奇正约数有多少个?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334586,7 +335867,10 @@ "id": "021040", "content": "已知集合$M=\\{-3,-2,-1,0,1,2\\}$, 点$P(a, b)$在直角坐标平面上, 且$a$、$b \\in M$.\\\\\n(1) 平面上共有多少个满足条件的点$P$?\\\\\n(2) 有多少个点$P$在第二象限内?\\\\\n(3) 有多少个点$P$不在函数$y=x$图像上?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334605,7 +335889,10 @@ "id": "021041", "content": "某班有男生$17$名, 女生$21$名, 从该班中选出学生代表参加校学代会.\\\\\n(1) 若学校分配给该班$1$名代表, 则可以有多少种不同的选取结果?\\\\\n(2) 若学校分配组该班$2$名代表, 且男、女生各$1$名, 则可以有多少种选取结果?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334624,7 +335911,10 @@ "id": "021042", "content": "用$0$、$6$、$8$这三个数字可组成多少个没有重复数字的整数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334643,7 +335933,10 @@ "id": "021043", "content": "用$0$到$9$这十个数字可组成多少个能被$5$整除的无重复数字的二位数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334662,7 +335955,10 @@ "id": "021044", "content": "某班的新年联欢会原定的$5$个节目已排成节目单, 开始演出前又增加了$2$个新节目, 如果将这两个新节目插入原节目单中, 那么有多少种不同的插法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334681,7 +335977,10 @@ "id": "021045", "content": "用$0$到$9$这十个数字, 可组成多少个没有重复数字的四位数的偶数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334700,7 +335999,10 @@ "id": "021046", "content": "有$A$、$B$、$C$、$D$、$E$五列火车停在某车站并行的$5$条火车轨道上, 如果快车$A$不能停在第$3$道上, 慢车$B$不能停在第$1$道上, 那么这五列火车的停车方法有多少种?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334719,7 +336021,10 @@ "id": "021047", "content": "某班级周一的课表要排入政治、语文、数学、物理、化学、体育共$6$门学科, 如果第一节课不排体育课, 最后一节课不排数学课, 那么共有多少种不同的排法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334738,7 +336043,10 @@ "id": "021048", "content": "用$0$到$5$这六个数字可组成无重复数字的四位数的偶数, 且这个偶数的百位、十位上都是奇数, 满足条件的数共有多少个?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334757,7 +336065,10 @@ "id": "021049", "content": "用$1$、$2$、$3$、$4$、$5$可组成多少个无重复数字且比$13245$大的五位数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334776,7 +336087,10 @@ "id": "021050", "content": "写出从$1,2,3,4$四个数字中, 任取两个数字所组成的所有两位数.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334795,7 +336109,10 @@ "id": "021051", "content": "从$15$件不同的礼品中取出$4$件分送给$4$个学生, 共有多少种不同的送法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334814,7 +336131,10 @@ "id": "021052", "content": "从$5$名运动员中选出$3$名参加乒乓球团体比赛, 并排定他们的出场顺序, 有多少种不同的方法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334833,7 +336153,10 @@ "id": "021053", "content": "从若干种不同的盆景中选出$2$种摆放在阳台的左右两侧, 如果想要有$30$种不同的选法, 那么最少要准备多少种不同的盆景?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334852,7 +336175,10 @@ "id": "021054", "content": "从$2,3,4,5,7,11$这六个数字中选出$2$个数字作为分子和分母, 共能组成多少个大小不同的分数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334871,7 +336197,10 @@ "id": "021055", "content": "从$6$名志愿者中选出$4$人分别从事翻译、导游、导购、保洁工作, 其中甲、乙两人不能从事翻译工作, 选派志愿者的方案共有多少种?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334890,7 +336219,10 @@ "id": "021056", "content": "求下列各式中$n$($n \\in \\mathbf{N}$, $n \\geq 1$)的值:\\\\\n(1) $\\mathrm{P}_{2 n}^3=11\\mathrm{P}_n^3$, $n=$\\blank{50};\\\\\n(2) $\\mathrm{P}_n^5+\\mathrm{P}_n^4=4\\mathrm{P}_n^3$, $n=$\\blank{50};\\\\\n(3) $\\mathrm{P}_n^3=n \\mathrm{P}_3^3$, $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334909,7 +336241,10 @@ "id": "021057", "content": "用$1$、$2$、$3$、$4$、$5$、$6$能组成多少个没有重复数字且大于$500$的三位数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334928,7 +336263,10 @@ "id": "021058", "content": "已知$\\mathrm{P}_{10}^m=10 \\times 9 \\times \\cdots \\times 5$, 求正整数$m$的值\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "填空题", "ans": "", "solution": "", @@ -334947,7 +336285,10 @@ "id": "021059", "content": "从$6$名学生中任选$3$人分别担任语文、数学、英语课代表, 其中学生甲不能担任数学课代表, 共有多少种不同的选法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334966,7 +336307,10 @@ "id": "021060", "content": "化简: $\\dfrac 1{2 !}+\\dfrac 2{3 !}+\\dfrac 3{4 !}+\\cdots+\\dfrac{n-1}{n !}$($n \\in \\mathbf{N}$, $n \\geq 2$).", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -334985,7 +336329,10 @@ "id": "021061", "content": "求证: $\\mathrm{P}_1^1+2\\mathrm{P}_2^2+3\\mathrm{P}_3^3+\\cdots+n \\mathrm{P}_n^n=\\mathrm{P}_{n+1}^{n+1}-1$($n \\in \\mathbf{N}$).", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335004,7 +336351,10 @@ "id": "021062", "content": "$5$名学生站成一排, 其中甲学生不能站在排头的不同站法有多少种?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335023,7 +336373,10 @@ "id": "021063", "content": "$4$名教师、 $3$名男生、 $2$名女生排成一排, 要求$3$名男生排在一起, $2$名女生排在一起, 共有多少种不同的排队方法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335042,7 +336395,10 @@ "id": "021064", "content": "用$0$、$1$、$2$、$3$、$4$、$5$这六个数字可以组成多少个没有重复数字的四位数的奇数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335061,7 +336417,10 @@ "id": "021065", "content": "用$0$到$9$这十个数可以组成多少个没有重复数字的四位数?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335080,7 +336439,10 @@ "id": "021066", "content": "已知甲、乙、丙等$7$人站成一排, 求分别按下列要求排队各有多少种不同的排法.\\\\\n(1) 甲乙都与丙相邻;\\\\\n(2) 甲乙之间有且只有$1$人.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335099,7 +336461,10 @@ "id": "021067", "content": "乒乓球队的$10$名队员中有$3$名主力队员, 派$5$名队员参加比赛, 其中, $3$名主力队员要安排在第一、三、五位置, 其余$7$名队员中的$2$名要安排在第二、四位置, 共有多少种不同的安排方法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335118,7 +336483,10 @@ "id": "021068", "content": "已知甲、乙、丙等$7$人站成一排, 分别按下列要求排队, 求满足条件的不同的排队数.\\\\\n(1) 甲乙都与丙相邻;\\\\\n(2) 甲乙丙两两互不相邻;\\\\\n(3) 甲在乙的左边(不一定相邻).", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335137,7 +336505,10 @@ "id": "021069", "content": "四对夫妇坐成一排照相.\\\\\n(1) 每对夫妇都不能隔开的排法有多少种?\\\\\n(2) 每对夫妇都不能隔开, 且同性别人的不相邻的排法有多少种?\\\\", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335156,7 +336527,10 @@ "id": "021070", "content": "甲、乙等九名同学排成一排, 甲、乙之间总是间隔$2$位同学, 共有多少种排法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335175,7 +336549,10 @@ "id": "021071", "content": "已知二次函数$y=a x^2+b x+c$, 集合$M=\\{-2,-1,0,1,2,3,4\\}$, $a$、$b$、$c \\in M$, 且$a$、$b$、$c$两两不相等, 满足条件的函数图像中, 过原点的有多少条?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335194,7 +336571,10 @@ "id": "021072", "content": "试确定下列问题是排列问题还是组合问题.\\\\\n(1) $3$本不同的书借给甲、乙、丙$3$名学生, 每人$1$本, 有多少种不同的借法?\\\\\n(2) 从$10$本书中任意取$5$本赠送给$1$名学生, 有多少种不同的送法?\\\\\n(3) 从$15$人中选$3$人去参加数学竞赛, 有多少种不同的选法?\\\\\n(4) 从$15$人中选$3$人分别参加数学、物理、化学竞赛, 有多少种不同的选法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335213,7 +336593,10 @@ "id": "021073", "content": "某项测试共有两组试题. 要求从第一组$10$个问题中选择$8$个, 从第二组$5$个问题中选择$4$个, 要完成这项测试有多少种不同的选择试题的方法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335232,7 +336615,10 @@ "id": "021074", "content": "已知$100$件产品中有$2$件次品, 如果从这些产品中任取$5$件, 那么其中恰好有$2$件次品的取法有多少种?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335251,7 +336637,10 @@ "id": "021075", "content": "某班级共有$25$名团员, 其中$10$名男团员, $15$名女团员.\\\\\n(1) 如果从中推选$2$名男团员和$3$名女团员参加团代会, 那么有多少种不同的推选方法?\\\\\n(2) 如果从中推选$2$名男团员和$3$名女团员组成团支部分别担任不同职务, 那么有多少种不同的推选方法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335270,7 +336659,10 @@ "id": "021076", "content": "以某个圆周上的$10$个点为顶点, 可以作多少个三角形?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335289,7 +336681,10 @@ "id": "021077", "content": "求下列各式中$n$($n \\in \\mathbf{N}$, $n \\geq 1$)的值:\\\\\n(1) $\\mathrm{C}_n^5+\\mathrm{C}_n^6=\\mathrm{C}_{n+1}^3$, $n=$\\blank{50};\\\\\n(2) $\\mathrm{C}_{n+1}^{n-1}=\\dfrac 1{16} \\mathrm{P}_{n+1}^3$, $n=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "填空题", "ans": "", "solution": "", @@ -335308,7 +336703,10 @@ "id": "021078", "content": "求证: $\\mathrm{C}_n^m=\\dfrac{m+1}{n+1} \\mathrm{C}_{n+1}^{m+1}$($n$、$m \\in \\mathbf{N}$, $n \\geq m \\geq 1$).", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335327,7 +336725,10 @@ "id": "021079", "content": "计算: $\\mathrm{C}_3^0+\\mathrm{C}_4^1+\\mathrm{C}_5^2+\\cdots+\\mathrm{C}_{10}^7$.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335346,7 +336747,10 @@ "id": "021080", "content": "从$8$名男运动员与$7$名女运动员中选出$5$名男运动员与$5$名女运动员组成一个运动队, 不同的选法共有多少种?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335365,7 +336769,10 @@ "id": "021081", "content": "要从$6$名男学生与$6$名女学生中选出$2$名男学生与$2$名女学生组成一个学习小组, 共有多少种不同的选法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335384,7 +336791,10 @@ "id": "021082", "content": "已知平面上共有$10$个点, 其中有$4$个点在一条直线上, 除此之外再没有三点共线, 以这$10$个点为顶点能组成多少不同的三角形?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335403,7 +336813,10 @@ "id": "021083", "content": "用一组$5$条平行线与另一组$4$条平行线共可围成多少个平行四边形?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335422,7 +336835,10 @@ "id": "021084", "content": "已知$\\dfrac{\\mathrm{C}_{2 n}^{n-1}}{\\mathrm{C}_{2 n-2}^n}=\\dfrac{56}{15}$, 求正整数$n$的值.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335441,7 +336857,10 @@ "id": "021085", "content": "从$3$本不同的语文书、 $4$本不同的数学书和$3$本不同的物理书中取出$4$本书, 且要求三种书都有共有多少种不同的取法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335460,7 +336879,10 @@ "id": "021086", "content": "从$5$名女学生和$4$名男学生中选出$4$人担任$4$种不同的工作, 且要求选出的$4$人中男女学生都有, 共有多少种不同的选法?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335479,7 +336901,10 @@ "id": "021087", "content": "已知集合$A$、$B$都含有$12$个元素, $A \\cap B$含有$4$个元素, 集合$C$含有$3$个元素, 且$C \\subset A \\cup B$, $C \\cap B \\neq \\varnothing$, 求满足条件的集合$C$的个数.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335498,7 +336923,10 @@ "id": "021088", "content": "某旅游团要从$8$个风景点中选$2$个风景点作为当天的旅游地, 求分别满足以下条件的选法的种数.\\\\\n(1) 甲乙风景中至少选一个;\\\\\n(2) 甲乙风景点中至多选一个;\\\\\n(3) 甲乙风景点中必须选一个, 而且只能选一个.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335517,7 +336945,10 @@ "id": "021089", "content": "设有$3$个不同的小球放入$3$个不同的盒子, 把球全部放入盒内, 一个盒内允许装多个球, 求分别满足下列条件的放法数:\\\\\n(1) 恰有一个盒子是空盒;\\\\\n(2) 允许有空盒;\\\\\n(3) 有一个盒内不少于$2$个球.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335536,7 +336967,10 @@ "id": "021090", "content": "现安排甲乙丙丁戊$5$名同学参加志愿者服务, 每人从事翻译、导游、礼仪、司机四项工作之一, 每项工作至少有一人参加. 甲、乙不会开车但能从事其他三项工作, 丙、丁、戊都能胜任四项工作, 则不同的安排方案数是多少?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335555,7 +336989,10 @@ "id": "021091", "content": "一个家庭要将$2$个男孩和$3$个女孩送到私立学校. 有$5$所男子学校、$8$所女子学校, 以及$3$所男女合校.如果每个孩子去不同的学校, 这个家庭为他们的孩子可以选择多少组不同的$5$所学校?", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335574,7 +337011,10 @@ "id": "021092", "content": "(1) 分别计算$\\mathrm{C}_2^0+\\mathrm{C}_2^1+\\mathrm{C}_2^2$和$\\mathrm{C}_3^0+\\mathrm{C}_3^1+\\mathrm{C}_3^2+\\mathrm{C}_3^3$;\\\\\n(2) 猜想$\\mathrm{C}_n^0+\\mathrm{C}_n^1+\\mathrm{C}_n^2+\\cdots+\\mathrm{C}_n^{n-1}+\\mathrm{C}_n^n$($n \\in \\mathbf{N}$, $n>0$)的值, 并证明你的结论.", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "排列组合" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335593,7 +337033,9 @@ "id": "021093", "content": "袋中有$10$个球, 分别记有号码$0$、$1$、$2$、$3$、$4$、$5$、$6$、$7$、$8$、$9$. 求下列事件的概率:\\\\\n(1) 任意取出$2$个球, 号码为$1$、$2$;\\\\\n(2) 任意取出$3$个球, 没有号码$3$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335612,7 +337054,9 @@ "id": "021094", "content": "掷两颗骰子, 求出现下列事件的概率.\\\\\n(1) 两颗骰子的点数之和等于$2$;\\\\\n(2) 两颗骰子的点数之和等于$3$;\\\\\n(3) 两颗骰子的点数之和等于$5$;\\\\\n(4) 两颗骰子的点数之和等于$7$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335631,7 +337075,9 @@ "id": "021095", "content": "已知某班有$38$名学生, 小李、小王、小张是该班的$3$名学生, 某次班会决定随机地挑选$3$名学生在会上发言, 求下列事件出现的概率.\\\\\n(1) 小李、小王、小张按此次序被选中;\\\\\n(2) 小李、小王、小张按任意次序被选中.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335650,7 +337096,9 @@ "id": "021096", "content": "某剧场将举办$8$场音乐会. 其中$2$场演奏莫扎特的作品, 小方对$8$场音乐会都很感兴趣, 难于选择, 最后决定用抽签的方法决定参加哪两场音乐会. 小方抽到两场都是莫扎特音乐会的概率是多少? 两场中$1$场是莫扎特音乐会的概率是多少?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335669,7 +337117,9 @@ "id": "021097", "content": "一部$4$卷的文集, 按任意次序放到书架上, 求各卷自左向右或自右向左的卷号为$1$、$2$、$3$、$4$的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335688,7 +337138,9 @@ "id": "021098", "content": "某种密码由$8$个数字组成, 且每个数字可以是$0$、$1$、$2$、$\\cdots$、$9$中的任意一个数, 求这种密码由完全不同的数字组成的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335707,7 +337159,9 @@ "id": "021099", "content": "一工厂生产的$10$个产品中有$9$个一等品、$1$个二等品. 现从这批产品中抽取$4$个, 求其中恰好有一个二等品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335726,7 +337180,9 @@ "id": "021100", "content": "掷一颗骰子, 求出现点数不小于$2$的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335745,7 +337201,9 @@ "id": "021101", "content": "已知$10$个产品中有$3$个次品, 从中任取$5$个, 求至少有一个次品的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335764,7 +337222,9 @@ "id": "021102", "content": "某城镇共有$10000$辆自行车, 牌照编号从$00001$到$10000$, 求在此城镇中偶然遇到的一辆自行车, 其牌照号码中有数字$8$的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335783,7 +337243,9 @@ "id": "021103", "content": "将$n$间房间分给$n$个人, 每个人都以相等的可能性进入每一间房间. 而且每间房间里的人数没有限制, 求不出现空房的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335802,7 +337264,9 @@ "id": "021104", "content": "把$10$本书随机地排在书架上, 求其中指定的$3$本书排在一起的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335821,7 +337285,9 @@ "id": "021105", "content": "某人有$5$把钥匙, 但只有一把能打开门, 他每次取一把钥匙尝试开门, 求试到第$3$把钥匙时才打开门的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335840,7 +337306,9 @@ "id": "021106", "content": "某次测验有$10$道备用试题, 甲同学在这$10$道题中能够答对$6$题, 现在备用试题中随机抽考$5$题, 规定答对$4$题或$5$题为优秀, 答对$3$题为及格.\n(1) 求甲同学获优秀的概率;\\\\\n(2) 求甲同学至少能够及格的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335859,7 +337327,9 @@ "id": "021107", "content": "某中学有十八个班级, 每班选出三个代表出席学生代表会议, 从$54$名代表中任选$18$名组成工作委员会, 分别求下列事件的概率:\\\\\n(1) 高一(1)班在工作委员会中有代表;\\\\\n(2) 每个班级在工作委员会中都有代表.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335878,7 +337348,9 @@ "id": "021108", "content": "在分别写有数字$0$、$1$、$2$、$3$、$4$、$5$、$6$、$7$、$8$、$9$的$10$张一样的卡片中随机抽取$1$张. 设事件$A$: 出现奇数, 事件$B$: 出现偶数, 事件$C$: 大于$4$. 写出下列事件对应的集合:\\\\ \n(1) $A$、$C$同时发生;\\\\ \n(2) $B$、$C$至少有一个发生;\\\\\n(3) $A$、$B$同时发生.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335897,7 +337369,9 @@ "id": "021109", "content": "掷一颗骰子. 设事件$A$: 落地时向上的点数是奇数, 事件$B$: 落地时向上的点数是偶数, 事件$C$: 落地时向上的点数是$3$的倍数, 事件$D$: 落地时向上的点数是 $4$. 则下列每对事件中, 不是互斥事件的为\\bracket{20}.\n\\fourch{$A$与$B$}{$B$与$C$}{$A$与$D$}{$C$与$D$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -335916,7 +337390,9 @@ "id": "021110", "content": "掷一枚均匀的骰子, 用$A$表示事件``向上的点数至少为$5$'', 则$\\overline A$指什么事件? $\\overline A$的对立事件是什么?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335935,7 +337411,9 @@ "id": "021111", "content": "从一副扑克牌(去掉大、小王, 共$52$张)中随机选取$1$张, 下列每组事件是否是互斥事件? 若是互斥事件, 那么是否互为对立事件?\\\\\n(1) $C$: 这张牌是红心, $D$: 这张牌是方块;\\\\ \n(2) $C$: 这张牌牌面是 2, 3, 4, 5, 6, 7, 8, 9, 10 之一, $D$: 这张牌牌面是 J, Q, K, A 之一;\\\\\n(3) $C$: 这张牌牌面是 2, 3, 4, 5, 6, 7 之一且为方块, $D$: 这张牌牌面是 8, 9, 10, J, Q, K, A之一且为方块.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335954,7 +337432,9 @@ "id": "021112", "content": "设$A$与$B$是同一样本空间中的两个事件, ``$A$与$B$相互对立''是``$A$与$B$互斥''的什么条件? 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335973,7 +337453,9 @@ "id": "021113", "content": "设$A$、$B$为两个事件, 试用$A$、$B$表示下列各事件:\\\\\n(1) $A$、$B$两个事件中至少有一个发生;\\\\ \n(2) $A$事件发生且$B$事件不发生;\\\\\n(3) $A$、$B$两个事件中恰有一个不发生.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -335992,7 +337474,9 @@ "id": "021114", "content": "已知事件$A$与$B$互斥, 判断$\\overline A$与$B$的关系, 以及$\\overline A$与$\\overline B$的关系, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336011,7 +337495,9 @@ "id": "021115", "content": "小明在书店随机地选一本书, 设$A$: 小明选的书是数学书, $B$: 小明选的书是中文版的书, $C$: 小明选的书是$2022$年或$2022$年以后出版的书.\\\\\n(1) 写出$A \\cap B \\cap \\overline C$表示的事件;\\\\\n(2) 在什么情况下, $A \\cap B \\cap C=A$成立? 简要说明理由;\\\\\n(3) 在什么情况下, $\\overline C \\subseteq B$成立? 简要说明理由;\\\\\n(4) $\\overline A=B$的含义是否是``书店中所有的数学书都不是中文版的''? 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336030,7 +337516,9 @@ "id": "021116", "content": "掷一颗骰子, 求出现下列事件的概率.\\\\\n(1) 点数不是素数;\\\\\n(2) 点数小于$5$或者为奇数;\\\\\n(3) 点数小于$3$或者大于$4$;\\\\\n(4) 点数是偶数或者素数.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336049,7 +337537,9 @@ "id": "021117", "content": "已知射手甲击中目标的概率为$0.80$, 射手乙击中目标的概率为$0.70$, 两人同时击中目标的概率为$0.56$, 用事件$A$表示``甲击中目标'', 事件$B$表示``乙击中目标''.\\\\\n(1) 判断$A$和$B$是否为互斥事件;\\\\\n(2) 求射手甲或射手乙击中目标的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336068,7 +337558,9 @@ "id": "021118", "content": "一个袋子里有$20$个白球、$40$个黑球和$10$个红球. 任意摸出一个球, 若此球为红球, 则得$3$分; 若此球为白球, 则得$2$分; 若此球为黑球, 则得$1$分.用$\\xi$表示摸出球的得分, 例如``$\\xi=1$''表示摸出球为黑色这一事件, $P(\\xi=1)$表示``$\\xi=1$''出现的概率, 填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 概率 &$P(\\xi=1)$&$P(\\xi=2)$&$P(\\xi=3) $&$P(\\xi \\neq 3)$&$P(\\xi=1$或 $2$) &$P(\\xi \\neq 2)$\\\\\n\\hline 概率值 & \\blank{30} & \\blank{30}& \\blank{30}& \\blank{30}& \\blank{30}&\\blank{30} \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336087,7 +337579,9 @@ "id": "021119", "content": "盒子里装有大小与质地相同的红球与白球, 从中任取$3$个球. 设事件$A$表示: ``$3$个球中有$1$个红球、$2$个白球'', 事件$B$表示``$3$个球中有$2$个红球、$1$个白球''. 已知$P(A)=\\dfrac 3{10}$, $P(B)=\\dfrac 12$. 求``$3$个球中既有红球又有白球''的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336106,7 +337600,9 @@ "id": "021120", "content": "已知事件$A$与$B$互斥, 它们都不发生的概率为$\\dfrac 25$, 且$P(A)=2P(B)$. 求$P(\\overline A)$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336125,7 +337621,9 @@ "id": "021121", "content": "袋中有大小与质地相同的$12$个小球, 分别为红球、黑球、黄球、绿球, 从中任取$1$个球, 得到红球的概率为$\\dfrac 13$, 得到黑球或黄球的概率是$\\dfrac 5{12}$, 得到黄球或绿球的概率是$\\dfrac 5{12}$. 试求得到黑球、黄球、绿球的概率各是多少.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336144,7 +337642,9 @@ "id": "021122", "content": "已知某次足球比赛中, 学校足球队赢的概率为$0.7$, 打平的概率为$0.2$, 求学校足球队不输的概率与输球的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336163,7 +337663,9 @@ "id": "021123", "content": "掷$3$枚骰子, 求所得点数之和不小于$5$的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336182,7 +337684,9 @@ "id": "021124", "content": "设$A$、$B$是同一样本空间中的两个事件, 探索$P(A \\cup B)$, $P(A)$, $P(B)$, $P(A \\cap B)$之间的等量关系, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336201,7 +337705,9 @@ "id": "021125", "content": "设$A$、$B$、$C$表示三个随机事件, 试用$A$、$B$、$C$的关系和运算分别表示下列各随机事件:\\\\\n(1) 三个事件中至少有两个发生;\\\\\n(2) 三个事件中不多于两个发生;\\\\\n(3) 三个事件中恰有两个发生.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336220,7 +337726,9 @@ "id": "021126", "content": "近代数学家掷硬币试验的一些结果列于下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline 试验者 & 掷硬币次数$n$& 正面出现次数$m$\\\\\n\\hline 德$\\cdot$摩尔根 & 2048 & 1061 \\\\\n\\hline 蒲丰 & 4040 & 2048 \\\\\n\\hline 皮尔逊 & 12000 & 6019 \\\\\n\\hline 皮尔逊 & 24000 & 12012 \\\\\n\\hline 维尼 & 30000 & 14994 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n分别求正面出现的频率, 并根据这些结果说一说前人所做掷硬币试验反映了怎样的规律.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336239,7 +337747,9 @@ "id": "021127", "content": "两台机床加工同样的零件, 第一台出现废品的经验概率是$0.03$, 第二台出现废品的经验概率是$0.02$, 加工出来的零件放在一起, 并且已知第一台加工的零件比第二台加工的零件多一倍, 求任意取出的零件是合格品的经验概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336258,7 +337768,9 @@ "id": "021128", "content": "一批零件中有$9$个合格品和$3$个废品, 安装机器时, 从这批零件中随机取出一个, 如果每次取出的成品不放回去, 分别求在取得第$1$件合格品以前已取出$x$件废品数的概率, $x=0,1,2,3$.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336277,7 +337789,9 @@ "id": "021129", "content": "已知血型为$O$、$A$、$B$、$AB$型的概率分别为$0.46$、$0.40$、$0.11$、$0.03$. 任意抽取一人, 求下列事件的概率:\\\\\n(1) 抽出人为$O$型血的概率;\\\\\n(2) 抽出人为$A$或$B$型血的概率;\\\\\n(3) 抽出人不是$AB$型血的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336296,7 +337810,9 @@ "id": "021130", "content": "某生物研究所进行某种淡水鱼的人工孵化实验, 在同一条件下的实验结果显示, $10000$个鱼卵能孵出约$8520$尾鱼苗.\\\\\n(1) 求这种鱼卵孵化的频率(经验概率);\\\\\n(2) 估计$30000$个这种鱼卵能孵出多少尾鱼苗?\\\\\n(3) 若要孵出$5000$尾鱼苗, 估计需准备这种鱼卵多少个?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336315,7 +337831,9 @@ "id": "021131", "content": "对一批西装进行了多次抽检, 并记录结果如下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 抽取件数 & 50 & 100 & 200 & 300 & 400 & 500 \\\\\n\\hline 检出次品的件数 & 5 & 6 & 5 & 6 & 8 & 10 \\\\\n\\hline 检出次品的频率 & & & & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 根据表中数据, 计算并填写每次抽检中检出次品的频率;\\\\\n(2) 从这批西装中任抽一件, 抽到次品的经验概率是多少?\\\\\n(3) 如果要销售$2000$件西装, 至少需额外准备多少件正品西装供买到次品的顾客调换?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336334,7 +337852,9 @@ "id": "021132", "content": "在某城市中共发行甲、乙、丙三种报纸, 在这个城市的居民中, 订甲报的人占总人数的$45 \\%$, 订乙报的人占$35 \\%$, 订丙报的人占$30 \\%$, 同时订甲乙两报的人占$10 \\%$, 同时订甲丙两报的人占$8 \\%$, 同时订乙丙两报的人占$5 \\%$, 同时订三种报纸的人占$3 \\%$, 求:\\\\\n(1) 只订甲报的人所占百分比;\\\\\n(2) 只订甲报或乙报的人所占百分比;\\\\\n(3) 只订一种报纸的人所占百分比;\\\\\n(4) 正好订两种报纸的人所占百分比;\\\\\n(5) 在所有订报纸的人中, 至少订一种报纸的人所占百分比;\\\\\n(6) 不订任何报纸的人所占百分比.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336353,7 +337873,9 @@ "id": "021133", "content": "某人有$4$把钥匙, 其中$2$把能打开门. 现随机地取$1$把钥匙开门.\\\\\n(1) 如果将不能开门的钥匙立即㧅掉, 求第二次才能开门的概率;\\\\ \n(2) 如果试过的钥匙不扔掉, 求第二次才能开门的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336372,7 +337894,9 @@ "id": "021134", "content": "如果$A$、$B$是独立事件, $\\overline A$、$\\overline B$分别是$A$、$B$的对立事件, 那么以下等式不一定成立的是\\bracket{20}.\n\\twoch{$P(A \\cap B)=P(A) P(B)$}{$P(\\overline A \\cap B)=P(\\overline A) P(B)$}{$P(A \\cup B)=P(A)+P(B)$}{$P(\\overline A \\cap \\overline B)=[1-P(A)][1-P(B)]$}", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336391,7 +337915,9 @@ "id": "021135", "content": "如果从甲口袋中摸出一个红球的概率是$\\dfrac 13$, 从已口袋中摸出一个红球的概率是$\\dfrac 12$, 求下面四个事件的概率:\\\\\n(1) $2$个球不都是红球;\\\\\n(2) $2$个球都是红球;\\\\\n(3) 至少有一个红球;\\\\\n(4) $2$个球中恰好有$1$个红球.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336410,7 +337936,9 @@ "id": "021136", "content": "设甲和乙两射手独立地射击统一目标, 他们的命中率分别为$0.95$和$0.90$. 求:\\\\\n(1) 在一次射击中, 目标被击中的概率;\\\\\n(2) 目标只被甲击中的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336429,7 +337957,9 @@ "id": "021137", "content": "把分奖金问题的三局两胜改为五局三胜, 问:\\\\\n(1) 在比分$2:0$的情况下, 怎样分奖金公平?\\\\\n(2) 在比分$1:0$的情况下, 怎样分奖金公平?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336448,7 +337978,9 @@ "id": "021138", "content": "一电路由电池$A$与两个并联的电池$B$、$C$串联而成. 设$A$、$B$、$C$损坏的概率分别为$0.3$, $0.2$, $0.2$, 求电路发生间断的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336467,7 +337999,9 @@ "id": "021139", "content": "设每一架飞机引擎在飞行中故障率为$1-p$, 且各引擎是否发生故障是独立的, 如果有至少$50\\%$的引擎能正常运行, 飞机就可以成功飞行. 问对于多大的$p$而言, $4$引擎飞机比$2$引擎飞机更安全?", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336486,7 +338020,9 @@ "id": "021140", "content": "三人分别独立解一道题, 已知甲做对这道题的概率是$\\dfrac 34$, 甲、丙两人都做错的概率是$\\dfrac 1{12}$, 乙、丙两人都做对的概率是$\\dfrac 14$. 求:\\\\\n(1) 乙、丙两人各自做对这道题的概率;\\\\\n(2) 甲、乙、丙三人中至少有两人做对这道题的概率.", "objs": [], - "tags": [], + "tags": [ + "第八单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336505,7 +338041,9 @@ "id": "021141", "content": "为了了解某校高二年级学生的视力情况, 从中抽查了$60$名学生的视力, 在这个问题中总体是\\bracket{20}.\n\\twoch{该校全体学生}{该校高二年级全体学生}{该校全体学生的视力}{该校高二年级全体学生的视力}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336524,7 +338062,9 @@ "id": "021142", "content": "为了了解某校高二年级学生的体重情况, 从中抽取了$50$名学生的体重进行分析. 在这项调查中, 样本是指\\bracket{20}.\n\\twoch{该校高二年级的全体学生}{该校高二年级的全体学生的体重}{被抽取的$50$名学生}{被抽取的$50$名学生的体重}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336543,7 +338083,9 @@ "id": "021143", "content": "在一项种子发芽的实验中, 抽取$2000$粒种子做试验, 这个问题中, $2000$粒种子发芽情况是\\bracket{20}.\n\\fourch{样本}{总体}{样本容量}{个体}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336562,7 +338104,9 @@ "id": "021144", "content": "为了考查某地区高中毕业生的数学学业水平考试情况, 从中抽查了$200$名考生的数学成绩, 在这个问题中, 下面说法错误的是\\bracket{20}.\n\\onech{总体是被抽查的$200$名考生}{个体是每一个考生的数学成绩}{样本是$200$名考生的数学成绩}{样本容量是$200$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336581,7 +338125,9 @@ "id": "021145", "content": "某厂首批共计生产了$200$根灯管, 为了检测它们的使用寿命, 从中抽取$10$根灯管进行加速寿命试验. 在这个抽样调查中, 下列说法正确的有\\blank{50}(填写相应的序号).\\\\\n\\textcircled{1} $200$根灯管是总体;\\\\\n\\textcircled{2} 每根灯管的使用寿命是个体;\\\\\n\\textcircled{3} $10$根灯管是样本的容量;\\\\\n\\textcircled{4} 若测得$10$根灯管的平均寿命是$5$万小时, 则说明该批次中所有灯管的寿命均为$5$万小时;\\\\\n\\textcircled{5} 为了准确了解该批灯管的使用寿命, 应当使用普查的方法进行调查.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336600,7 +338146,9 @@ "id": "021146", "content": "研究上海市常住人口的年龄构成时, 根据$2020$年上海市第七次人口普查的结果, 全市常住人口数为$2487$万, 研究员随机抽样调查了$24000$个常住居民的年龄. 在这个抽样调查中, 总体和样本分别是什么? 样本量是多少?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336619,7 +338167,9 @@ "id": "021147", "content": "完成下列任务所获得的数据是观测数据还是实验数据?\\\\\n(1) 某杂志对不同年龄层的阅读兴趣进行了调查, 最终揭晓``文学、旅游、医用卫生''这三大类图书受到各个年龄层次的喜爱;\\\\\n(2) 我国每十年进行一次全国人口普查, 根据$2020$年上海市第七次人口普查的结果, 全市常住人口数为$2487$万;\\\\\n(3) 某疫苗实验中, 接种该疫苗的人员重症率为$0.8 \\%$, 末接种该疫苗的人员重症率为$5.2 \\%$.\\\\\n其中, (1)是\\blank{50}数据; (2)是\\blank{50}数据; (3)是\\blank{50}数据.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336638,7 +338188,9 @@ "id": "021148", "content": "高二学生小明想要调研上海市高中生线上学习期间的日均运动时长, 为此他制作了电子调查问卷, 并通过班级群、朋友圈等网络渠道进行发放, 并回收得到$64$份有效问卷. 根据回收的数据, 小明在计算后推断: 上海市高中生线上学习期间的日均运动时长为$0.7$小时. 你认为小明的推断可靠么? 为什么?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336657,7 +338209,9 @@ "id": "021149", "content": "数据:$-2,-1,0,1,2$的方差是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336676,7 +338230,9 @@ "id": "021150", "content": "一组两两不同的数据从小到大排列后, 如果第$61$个数和第$62$个数的平均数是这组数据的中位数, 则这组数据的容量为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336695,7 +338251,9 @@ "id": "021151", "content": "已知总体的各个体的值由小到大依次为$2,3,3,7, a, b, 12,13.7,18.3,20$, 且总体的中位数为$10.5$, 若要使该总体的方差最小, 则$a$、$b$的取值分别是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336714,7 +338272,9 @@ "id": "021152", "content": "有一组数据$x_1, x_2, x_3, \\cdots, x_n$, 其平均值$\\overline x=5$, 方差$s^2=3$. 已知$y_i=3 x_i+2$($i=1,2,3, \\ldots, n$), 求$y_1, y_2, y_3, \\cdots, y_n$的平均值与方差.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336733,7 +338293,9 @@ "id": "021153", "content": "某计算机操作培训班各学员的考试成绩如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline 得分 & 100 & 90 & 80 & 70 & 67 & 65 & 63 & 55 \\\\\n\\hline 人数 & 2 & 3 & 10 & 25 & 13 & 3 & 2 & 2 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n求学员考试成绩的平均数、中位数和得分的方差. (结果精确到$0.1$)", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336752,7 +338314,9 @@ "id": "021154", "content": "如果采用随机抽样, 从容量为$N$的总体中抽取一个容量为$n$的样本, 那么每个个体被抽中的概率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336771,7 +338335,9 @@ "id": "021155", "content": "下列抽样实验中, 最适合用抽签法的是\\bracket{20}.\n\\onech{从某厂生产的$30000$件产品中抽取$600$件进行质量检验}{从某班级$30$名学生中抽取$5$名学生参加一个座谈会}{从某校高中一年级$320$名学生中抽取$30$名学生了解体育运动情况}{从某厂生产的$30000$件产品中抽取$10$件进行质量检验}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336790,7 +338356,9 @@ "id": "021156", "content": "将一个总体分为$A$、$B$、$C$三层, 各层个体数之比为$5: 3: 2$, 若用分层抽样抽取方法容量为$100$的样本, 则应从$C$中抽取\\blank{50}个个体.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336809,7 +338377,9 @@ "id": "021157", "content": "某商场有四类食品, 其中粮食类、植物油类、动物性食品类及果蔬类分别有$40$种、$10$种、$30$种、$20$种, 现从中抽取一个容量为$20$的样本进行食品安全检测, 若采用分层抽样的方法抽取样本, 则抽取的植物油类与果蔬类食品种类之和是\\bracket{20}.\n\\fourch{$4$}{$5$}{$6$}{$7$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336828,7 +338398,9 @@ "id": "021158", "content": "一工厂一共生产了某种产品$13400$件, 它们来自甲、乙、丙$3$条生产线. 为检查这批产品的质量, 决定采用分层抽样的方法进行抽样. 已知从甲、乙、丙$3$条生产线抽取的个体数刚好组成一个等差数列, 则乙生产线一共生产了\\blank{50}件产品.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336847,7 +338419,9 @@ "id": "021159", "content": "下列问题中, 采用怎样的抽样方法较为合理?\\\\\n(1) 从$20$台手提电脑中抽取$4$台进行质量检查;\\\\\n(2) 某学校共有七个年级$1600$名学生, 其中, 六年级学生$160$名, 七年级学生$160$名, 八年级学生$240$名, 九年级学生$240$名, 高中一年级学生$200$名, 高中二年级学生$280$名, 高中三年级学生$320$名, 从中抽取一个容量为$160$的样本.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336866,7 +338440,9 @@ "id": "021160", "content": "借助课本第$174$页附录``随机数表'', 可以用随机数表法从$1$到$50$的数字中随机抽取$10$个数字作样本. 假设从第$4$行第一个数字``$4$''开始读数, 请写出所抽取的样本.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336885,7 +338461,9 @@ "id": "021161", "content": "某校有学生$2100$人, 其中高二学生$700$人, 高三学生$620$人, 其余均为高三学生. 为了了解学生的身体素质情况, 现采用按年级分层抽样的方法从中抽取部分学生进行调查. 已知在高一学生中的抽样人数是$34$, 试计算本次抽样的样本容量, 以及在高二学生中的抽样人数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336904,7 +338482,9 @@ "id": "021162", "content": "某学校共有$2000$名学生, 从中选取$20$名学生参加学生代表大会, 试采用随机抽样和分层抽样两种方法进行具体实施.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336923,7 +338503,9 @@ "id": "021163", "content": "某学校学生志愿者协会共有$250$名成员, 其中高中一年级学生$88$名, 高中二年级学生$112$名, 高中三年级学生$50$名, 为了了解志愿者活动与学校学习之间的关系, 需要抽取$50$名学生进行调查, 试确定抽取方法并写出过程.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -336942,7 +338524,9 @@ "id": "021164", "content": "下列关于频率分布直方图的说法中, 正确的是\\bracket{20}.\n\\onech{频率分布直方图中矩形的高表示该组个体在样本中出现的频率}{组距越小, 频率分布直方图在呈现数据分布情况时越有效}{在频率分布直方图中, 所有小矩形的面积之和为$1$}{频率分布表比频率分布直方图更能直观形象地反映样本数据的分布规律}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -336961,7 +338545,9 @@ "id": "021165", "content": "某项比赛中有$9$位裁判打分, 去掉一个最高分, 去掉一个最低分后, 剩余$7$个分数的平均分即为最终得分. 小明在这项比赛中的最终得分为$91$分, 这$9$个得分的茎叶图如图所示, 其中$1$个数据模糊, 无法辨认, 在图中以$x$表示: 则$x=$\\blank{50}.\n\\begin{center}\n\\begin{tabular}{c|cccccc}\n8 & 7 & 7 \\\\\n9 & 4 & 0 & 1 & 0 & $x$ & 1\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336980,7 +338566,9 @@ "id": "021166", "content": "如图为样本容量为$200$的频率分布直方图, 则样本数据落在$[6,10)$内的频率为\\blank{50}, 样本数据落在$[10,18)$的频数为\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.2, yscale = 30]\n\\draw [->] (0,0) -- (24,0) node [below right] {样本数据};\n\\draw [->] (0,0) -- (0,0.12) node [left] {频率/组距};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {2/0.02,6/0.08,10/0.09,14/0.03,18/0.03}\n{ \\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (4,0) --++ (0,{-\\j});};\n\\draw (22,0) node [below] {$22$};\n\\draw [dashed] (2,0.02) -- (0,0.02) node [left] {$0.02$};\n\\draw [dashed] (14,0.03) -- (0,0.03) node [left] {$0.03$};\n\\draw [dashed] (6,0.08) -- (0,0.08) node [left] {$0.08$};\n\\draw [dashed] (10,0.09) -- (0,0.09) node [left] {$0.09$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -336999,7 +338587,9 @@ "id": "021167", "content": "交通部门计划对某路段进行限速, 为调查限速$60$公里/时是否合理, 对通过该路段的$500$辆汽车进行检测, 将获得的数据按$[40,50),[50,60),[60,70),[70,80]$分组, 绘制出下面的频率分布直方图. 则这$500$辆且车中车速低于限速的汽车有\\blank{50}辆.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, xscale = 0.05, yscale = 60]\n\\draw [->] (0,0) -- (90,0) node [below right] {车速};\n\\draw [->] (0,0) -- (0,0.05) node [left] {频率/组距};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i/\\j in {40/0.025,50/0.035,60/0.03,70/0.01}\n{\\draw (\\i,0) node [below] {$\\i$} --++ (0,\\j) --++ (10,0) --++ (0,{-\\j});};\n\\draw (80,0) node [below] {$80$};\n\\foreach \\i/\\j/\\k in {0.01/0.01/70,0.025/a/40,0.03/0.03/60,0.035/0.035/50}\n{\\draw [dashed] (\\k,\\i) -- (0,\\i) node [left] {$\\j$};};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337018,7 +338608,9 @@ "id": "021168", "content": "为统计某校高二年级学生的身体健康状况, 选取该校男女生各$20$名的体重数据, 绘制如下男生、女生体重分布茎叶图:\n\\begin{center}\n\\begin{tabular}{ccccccccccc|c|cccccc}\n&&&&&&&&&& 女 & & 男\\\\\n&&&&&&&&& 5 & 7 & 4 & 6 & 8\\\\\n0 & 0 & 1 & 3 & 4 & 4 & 6 & 6 & 7 & 7 & 9 & 5 & 2 & 4 & 5 & 5 & 7 & 8\\\\\n&&&&&0 & 0 & 1 & 3 & 6 & 8 & 6 & 3 & 3 & 5 & 7 & 9\\\\\n&&&&&&&&&& 3 & 7 & 0 & 2 & 2 & 7\\\\\n&&&&&&&&&&&8 & 6 & 8 & 9\n\\end{tabular}\n\\end{center}\n由图可知, 女生的体重分布集中程度较\\blank{20}, 男生的体重分布集中程度较\\blank{20}. 经计算，女生体重样本的标准差为\\blank{50}, 男生体重样本的标准差为\\blank{50}, 与前述结论一致.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337037,7 +338629,9 @@ "id": "021169", "content": "某工厂制造了一批零件, 研究它们的质量(单位: g), 样本数据如下所示:\n\\begin{center}\n$10.3$, $9.6$, $8.9$, $9.4$, $10.1$, $11.2$, $10.0$, $9.7$, $10.1$, $10.5$, $10.0$, $9.3$, $11.0$, $10.8$\n\\end{center}\n为了更直观地观测这些零件的质量分布, 请绘制茎叶图, 并对这些分析其质量分布.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337056,7 +338650,9 @@ "id": "021170", "content": "某校教师进行体格检查, 测得他们的收缩压(血压, 单位: 毫米汞柱)的值如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline 收缩压范围 &$89.5 \\sim 104.4$&$104.5 \\sim 119.4$&$119.5 \\sim 134.4$&$134.5 \\sim 149.4$&$149.5 \\sim 164.4$&$164.5 \\sim 179.4$\\\\\n\\hline 人数 & 24 & 62 & 72 & 26 & 12 & 4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n请根据表中数据, 绘制该校教师收缩压分布的频率直方图和频率折线图.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337075,7 +338671,9 @@ "id": "021171", "content": "为考察某校高中三年级男学生的身高, 随机地抽取$40$名男学生, 测得他们的身高 (单位: $\\text{cm}$) 如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\\hline 166 & 176 & 170 & 173 & 176 & 165 & 168 & 181 & 178 & 163 \\\\\n\\hline 174 & 170 & 189 & 178 & 170 & 171 & 175 & 182 & 167 & 180 \\\\\n\\hline 167 & 174 & 168 & 172 & 177 & 178 & 177 & 173 & 177 & 164 \\\\\n\\hline 177 & 175 & 187 & 179 & 172 & 166 & 174 & 165 & 170 & 170 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 填写下表:\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|}\n\\hline 身高 & {$[162,166)$} & {$[166,170)$} & {$[170,174)$} & {$[174,178)$} & {$[178,182)$} & {$[182,186)$} & {$[186,190]$} \\\\\n\\hline 频数 & & & & & & & \\\\\n\\hline 频率 & & & & & & & \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(2) 请计算样本的极差与平均值;\\\\\n(3) 画出样本的频率直方图, 分析该校高中三年级学生身高的大致分布情况.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337094,7 +338692,9 @@ "id": "021172", "content": "若样本平均数为$\\overline x$, 总体平均数为$a$, 则\\bracket{20}.\n\\fourch{$\\overline x=a$}{$\\overline x$是$a$的近似值}{$a$是$\\overline x$的估计值}{$\\overline x$是$a$的估计值}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -337113,7 +338713,9 @@ "id": "021173", "content": "若$\\sum_{i=1}^{20} x_i=18$, $\\sum_{i=1}^{20} y_i=7$, 则$\\sum_{i=1}^{20}(2 x_i-3 y_i+1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337132,7 +338734,9 @@ "id": "021174", "content": "若$x_1, x_2, \\cdots, x_n$的标准差为$4$, 则$3 x_1-2,3 x_2-2, \\cdots, 3 x_n-2$的标准差为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337151,7 +338755,9 @@ "id": "021175", "content": "甲、乙两人在相同的条件下各射靶$10$次, 命中的环数如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 甲 & 7 & 8 & 6 & 8 & 6 & 5 & 9 & 10 & 7 & 4 \\\\\n\\hline 乙 & 9 & 5 & 7 & 8 & 7 & 6 & 8 & 6 & 7 & 6 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n则两人谁射靶更准? 谁发挥更稳定? 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337170,7 +338776,9 @@ "id": "021176", "content": "从一个有$800$户居民的小区中随机抽取一个$30$户的样本, 样本中每户的人数如表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 5 & 6 & 3 & 3 & 2 & 3 & 3 & 3 & 4 & 4 & 3 & 2 & 7 & 4 & 3 \\\\\n\\hline 5 & 4 & 4 & 3 & 3 & 4 & 3 & 3 & 1 & 2 & 4 & 3 & 4 & 2 & 4 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n(1) 估计该小区平均每户的人数;\\\\\n(2) 估计该小区居民总数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337189,7 +338797,9 @@ "id": "021177", "content": "在发生某公共卫生事件期间, 有专业机构认为该事件在一定时间内没有发生大规模群体感染的标志为``连续$10$天, 每天新增疑似病例不超过$7$人''. 根据过去$10$天甲、乙、丙、丁四地新增疑似病例数据, 一定符合该标志的是\\bracket{20}.\n\\twoch{甲地: 总体均值为$3$, 中位数为$4$}{乙地: 中位数为$2$, 众数为$3$}{丙地: 总体均值为$2$, 总体方差为$3$}{丁地: 总体均值为$1$, 总体方差大于$0$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -337208,7 +338818,9 @@ "id": "021178", "content": "从某中学$200$名新生中随机抽取$10$名进行身高测量, 得数据为:\n168、159、166、163、170、161、167、155、162、169(单位: $\\text{cm}$).\n试估计该中学$200$名新生的平均身高以及身高高于$165 \\text{cm}$的人数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337227,7 +338839,9 @@ "id": "021179", "content": "关于百分位数, 下列说法中正确的是\\blank{50}. (填入所有正确的编号)\\\\\n\\textcircled{1} 百分位数将样本数据等分为$100$份;\\\\\n\\textcircled{2} 一组数据中不同的百分位数可能相等;\\\\\n\\textcircled{3} 恰好有$k \\%$的数据比第$k$百分位数小;\\\\\n\\textcircled{4} 某样本数据的第$k$百分位数即是总体数据的第$k$百分位数.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337246,7 +338860,9 @@ "id": "021180", "content": "已知$3.2$、$3.4$、$3.8$、$4.2$、$4.3$、$4.5$、$6.6$与$x$这八个数据的第$65$百分位数是$4.5$, 则实数$x$的取值范围是\\bracket{20}.\n\\fourch{$[4.5,+\\infty)$}{$[4.5,6.6)$}{$(4.5,+\\infty)$}{$(4.5,6.6]$}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -337265,7 +338881,9 @@ "id": "021181", "content": "某旅游网考察了一个景点附近的 15 家酒店, 根据清洁程度、舒适程度、服务态度、位置、早餐质量等给每个酒店一个评分(满分为 10 分), 评分结果如下:\n9.9, 8.8, 8.7, 9.1, 8.2, 8.5, 9.7, 8.9, 9.5, 9.8, 9.2, 8.9, 9.6, 8.6, 8.9.\\\\\n(1) 计算第三个四分位数;\\\\\n(2) 某网站计划将评分在第$95$百分位数以上的酒店列为优先推荐酒店. 若某酒店的评分为$9.6$, 该酒店能否列为优先推荐酒店?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337284,7 +338902,9 @@ "id": "021182", "content": "某校有甲、乙两个数学兴趣班, 其中甲班与乙班人数之比为$4: 5$. 在某一次考试中, 两个兴趣班全体学生的平均成绩为$88$分, 甲班的平均成绩是$86$分, 方差为$10$, 乙班的方差为$8$. 试用这些数据求乙班的平均成绩, 并计算两个班级学生成绩的总体方差. (精确到$0.01$)", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337303,7 +338923,9 @@ "id": "021183", "content": "点$P(1,1)$与椭圆$\\dfrac{x^2}2+y^2=1$的位置关系是\\bracket{20}.\n\\fourch{点在椭圆内}{点在椭圆上}{点在椭圆外}{无法确定}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -337322,7 +338944,9 @@ "id": "021184", "content": "若椭圆$\\dfrac{x^2}2+y^2=1$与直线$y=x+m$有两个公共点, 则实数$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337341,7 +338965,9 @@ "id": "021185", "content": "若曲线$\\dfrac{x^2}2+y^2=1$($y \\geq 0$)与直线$y=x+m$有两个公共点, 则实数$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337360,7 +338986,9 @@ "id": "021186", "content": "如果直线$y=k x+1$与椭圆$\\dfrac{x^2}5+\\dfrac{y^2}m=1$恒有公共点, 那么实数$m$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337379,7 +339007,9 @@ "id": "021187", "content": "若直线$y=2 x$与椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的一个交点的横坐标恰为$c$($c$为半焦距), 则$\\dfrac ca$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337398,7 +339028,9 @@ "id": "021188", "content": "已知点$A(6,8)$, 点$P$是圆$x^2+y^2=16$上的动点, 则线段$AP$的中点$M$的轨迹方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337417,7 +339049,9 @@ "id": "021189", "content": "椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$上一点$M$到左焦点$F_1$的距离为$2, N$是$MF_1$的中点, $O$是坐标原点, 求$|ON|$的长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337436,7 +339070,9 @@ "id": "021190", "content": "已知椭圆$C$的焦点分别为$F_1(-2 \\sqrt 2, 0)$、$F_2(2 \\sqrt 2, 0)$, 长轴长为$6$, 直线$y=x+2$交椭圆$C$于$A$、$B$两点, 求线段$AB$的中点的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337455,7 +339091,9 @@ "id": "021191", "content": "已知椭圆$\\dfrac{y^2}9+x^2=1$, 一条不与坐标轴平行的直线$l$与该椭圆交于不同的两点$M$、$N$, 且线段$MN$的中点的横坐标为$-\\dfrac 12$, 求直线$l$的斜率的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337474,7 +339112,9 @@ "id": "021192", "content": "已知椭圆$\\dfrac{x^2}2+y^2=1$. 求过点$M(\\dfrac 12, \\dfrac 12)$且被$M$平分的弦所在直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337493,7 +339133,9 @@ "id": "021193", "content": "已知椭圆$\\dfrac{x^2}2+y^2=1$. 求斜率为$2$的平行弦中点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337512,7 +339154,9 @@ "id": "021194", "content": "已知椭圆$\\dfrac{x^2}2+y^2=1$. 过椭圆的左焦点$F_1$引椭圆的割线, 求截得的弦的中点$Q$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337531,7 +339175,9 @@ "id": "021195", "content": "椭圆$\\dfrac{y^2}{100}+\\dfrac{x^2}{64}=1$上任一点到两焦点的距离和为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337550,7 +339196,9 @@ "id": "021196", "content": "与椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$有相同的焦点, 且经过点$(3,-2)$的椭圆方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337569,7 +339217,9 @@ "id": "021197", "content": "过椭圆$\\dfrac{x^2}{169}+\\dfrac{y^2}{144}=1$的右焦点且与圆$x^2+y^2=1$相切的直线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337588,7 +339238,9 @@ "id": "021198", "content": "若椭圆$\\dfrac{x^2}9+\\dfrac{y^2}4=1$的弦$AB$被点$P(1,1)$平分, 则$AB$所在直线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337607,7 +339259,9 @@ "id": "021199", "content": "直线$y=k x-2$交椭圆$\\dfrac{x^2}{80}+\\dfrac{y^2}{20}=1$于不同两点$P$和$Q$, 若$PQ$中点横坐标等于 2 , 求弦长$|PQ|$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337626,7 +339280,9 @@ "id": "021200", "content": "已知椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)与直线$x+2 y-2=0$交于$A$、$B$两点, $|AB|=\\sqrt 5$, 且$AB$的中点的坐标为$(m, \\dfrac 12)$, 求此椭圆的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337645,7 +339301,9 @@ "id": "021201", "content": "已知椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$的焦点为$F_1$、$F_2$, 点$P$为其上的动点, 当$\\angle F_1PF_2$为钝角时, 求点$P$的横坐标的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337664,7 +339322,9 @@ "id": "021202", "content": "已知椭圆$\\dfrac{x^2}4+\\dfrac{y^2}3=1$上有不同的两点关于直线$y=4 x+m$对称, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337683,7 +339343,9 @@ "id": "021203", "content": "椭圆$C: \\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的两个焦点为$F_1$、$F_2$, 点$P$在椭圆$C$上, 且$PF_1 \\perp F_1F_2$, $|PF_1|=\\dfrac 43$, $|PF_2|=\\dfrac{14}3$.\\\\\n(1) 求椭圆$C$的方程;\\\\\n(2) 若直线$l$过圆$x^2+y^2+4 x-2 y=0$的圆心$M$, 交椭圆$C$于$A$、$B$两点, 且$A$、$B$两点关于$M$对称, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337702,7 +339364,9 @@ "id": "021204", "content": "若椭圆的两个焦点三等分长轴$AA_1, B$为短轴的一个端点, 则$\\angle BAA_1$的正切值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337721,7 +339385,9 @@ "id": "021205", "content": "椭圆$x^2+2 y^2=98$上的点到点$P(0,5)$的距离的最大值是\\blank{50}, 最小值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337740,7 +339406,9 @@ "id": "021206", "content": "中心在原点, 一个焦点是$(0,5 \\sqrt 2)$, 截直线$y=3 x-2$所得弦的中点的横坐标为$\\dfrac 12$的椭圆方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337759,7 +339427,9 @@ "id": "021207", "content": "椭圆短轴的一个端点与两个焦点构成一个正三角形, 焦点到椭圆上的点的最短距离是$\\sqrt 3$, 则椭圆的标准方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337778,7 +339448,9 @@ "id": "021208", "content": "已知$M(x, y)$是椭圆$\\dfrac{x^2}9+\\dfrac{y^2}{25}=1$上的动点, 则$x-y$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337797,7 +339469,9 @@ "id": "021209", "content": "若$P$为椭圆$\\dfrac{x^2}3+\\dfrac{y^2}4=1$上任一点, $F_1$、$F_2$为其两焦点, 则$\\angle F_1PF_2$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337816,7 +339490,9 @@ "id": "021210", "content": "若$AB$为过椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)中心的弦, $F_1$为此椭圆的焦点, 则$\\triangle F_1AB$的面积的最大值是\\bracket{20}.\n\\fourch{$a c$}{$a b$}{$b c$}{$b^2$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -337835,7 +339511,9 @@ "id": "021211", "content": "已知$F_1$、$F_2$分别为椭圆$\\dfrac{x^2}{a^2}+\\dfrac{y^2}{b^2}=1$($a>b>0$)的左、右焦点, 点$P$在椭圆上, $O$为坐标原点, $\\triangle POF_2$是面积为$4 \\sqrt 3$的正三角形, 求$b^2$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337854,7 +339532,9 @@ "id": "021212", "content": "在椭圆$\\dfrac{x^2}{100}+\\dfrac{y^2}{25}=1$上找一点$P$, 使它到直线$l: 3 x+8 y+72=0$的距离最小, 并求此最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337873,7 +339553,9 @@ "id": "021213", "content": "过椭圆$C: x^2+2 y^2=2$的右焦点的直线$l$交椭圆于$A, B$两点, 且$\\triangle AOB$的面积是$\\dfrac 49$, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337892,7 +339574,9 @@ "id": "021214", "content": "过点$P(-\\sqrt 3, 0)$作直线$l$交椭圆$11 x^2+y^2=9$于$M$、$N$两点, 当直线$l$的倾斜角为何值时, 以$MN$为直径的圆恰好通过椭圆中心?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -337911,7 +339595,9 @@ "id": "021215", "content": "若$P$是双曲线$x^2-y^2=9$的左支上一点, $F_1$、$F_2$分别是双曲线的左、右焦点, 则$|PF_1|-|PF_2|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337930,7 +339616,9 @@ "id": "021216", "content": "设$a$是正实数. 已知点$P$与$A(-a, 0)$、$B(a, 0)$两定点的连线的斜率之积为定值$t(t>0)$, 则点$P$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337949,7 +339637,9 @@ "id": "021217", "content": "若椭圆$\\dfrac{x^2}4+\\dfrac{y^2}{2 k^2}=1$与双曲线$\\dfrac{x^2}{k^2}-\\dfrac{y^2}2=1$的焦点相同, 则正数$k$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337968,7 +339658,9 @@ "id": "021218", "content": "若椭圆$\\dfrac{x^2}4+\\dfrac{y^2}{2 k^2}=1$与双曲线$\\dfrac{x^2}{k^2}-\\dfrac{y^2}2=1$的焦距相同, 则正数$k$的值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -337987,7 +339679,9 @@ "id": "021219", "content": "已知$F_1(0,-5)$、$F_2(0,5)$两点, 若动点$P$满足$|PF_1|-|PF_2|=8$, 则点$P$的轨迹是\\bracket{20}.\n\\twoch{双曲线}{双曲线靠近$F_1$的一支}{双曲线靠近$F_2$的一支}{一条射线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338006,7 +339700,9 @@ "id": "021220", "content": "在$\\triangle ABC$中, 已知$A(-4,0)$、$B(4,0)$两点. 若$\\sin A-\\sin B=\\dfrac 12 \\sin C$, 则顶点$C$的轨迹方程是\\bracket{20}.\n\\twoch{$\\dfrac{x^2}4-\\dfrac{y^2}{12}=1$($x<-2$)}{$\\dfrac{x^2}4-\\dfrac{y^2}{12}=1$($x>2$)}{$\\dfrac{x^2}{12}-\\dfrac{y^2}4=1$($x>2 \\sqrt 3$)}{$\\dfrac{x^2}{12}-\\dfrac{y^2}4=1$($y \\neq 0$)}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338025,7 +339721,9 @@ "id": "021221", "content": "已知椭圆$\\dfrac{x^2}{m^2}+\\dfrac{y^2}{n^2}=1$($|m|>|n|$)和双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$相同的两焦点$F_1$、$F_2$, 若点$P$为两曲线的一个交点, 则$|PF_1|\\cdot|PF_2|$等于\\bracket{20}.\n\\fourch{$m^2-a^2$}{$\\dfrac 12(m^2-a^2)$}{$m-n$}{$a^2-m^2$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338044,7 +339742,9 @@ "id": "021222", "content": "写出分别满足下列条件的双曲线的标准方程.\\\\\n(1) 曲线上的点$P$到点$F_1(4,0)$的距离与它到点$F_2(-4,0)$的距离的差的绝对值等于$6$;\\\\\n(2) 曲线上的点$P$到点$F_1(0,-10)$的距离与它到点$F_2(0,10)$的距离的差等于$16$;\\\\\n(3) 焦点在$x$轴上, 且双曲线经过点$(-\\sqrt 2,-\\sqrt 3)$、$(\\dfrac{\\sqrt {15}}3, \\sqrt 2)$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338063,7 +339763,9 @@ "id": "021223", "content": "已知双曲线的对称轴为坐标轴, 焦点为$(-6,0)$、$(6,0)$, 且双曲线经过点$(-5,2)$, 求此双曲线的标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338082,7 +339784,9 @@ "id": "021224", "content": "已知双曲线的焦点在坐标轴上且关于原点对称, 其焦距椭圆$\\dfrac{x^2}{144}+\\dfrac{y^2}{169}=1$的焦距相同, 且该双曲线经过点$(3,4 \\sqrt 2)$, 求它的标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338101,7 +339805,9 @@ "id": "021225", "content": "给出问题: $F_1$、$F_2$是双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}{20}=1$的两个焦点, 点$P$在此双曲线上, 且点$P$到点$F_1$的距离等于$9$, 求点$P$到点$F_2$的距离. 某学生的解答如下: 双曲线的实轴长为$8$, 由$|PF_1|-|PF_2|=8$, 得$|PF_2|=1$或$17$, 该学生的解答是否正确? 若正确, 说明他的解题依据; 若不正确, 请你写出正确的结果.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338120,7 +339826,9 @@ "id": "021226", "content": "在相距$1000$米的$F_1$、$F_2$两地听到炮声的时间差是$2$秒 (声速: $340$米/秒). 若以$F_1$和$F_2$的中点为原点, $F_1$与$F_2$两地所在直线为$x$轴, 建立平面直角坐标系, 则炮的位置所在曲线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338139,7 +339847,9 @@ "id": "021227", "content": "若方程$\\dfrac{x^2}{3-\\lambda}+\\dfrac{y^2}{\\lambda-1}=1$表示双曲线, 则$\\lambda$的取值范围是\\bracket{20}.\n\\fourch{$1<\\lambda<3$}{$\\lambda<1$}{$\\lambda>3$}{$\\lambda<1$或$\\lambda>3$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338158,7 +339868,9 @@ "id": "021228", "content": "设$m, n$是实数. 若方程$\\dfrac{x^2}{m^2+n}-\\dfrac{y^2}{2 m^2-n}=1$表示双曲线, 且该双曲线的焦距为$6$, 则$n$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338177,7 +339889,9 @@ "id": "021229", "content": "设方程$\\dfrac{x^2}{m+2}-\\dfrac{y^2}{m+1}=1$表示焦点在$y$轴上的双曲线, 求实数$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338196,7 +339910,9 @@ "id": "021230", "content": "过双曲线$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$的右焦点$F_2$作$x$轴的垂线, 求此垂线与双曲线的交点$M$到左焦点$F_1$的距离.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338215,7 +339931,9 @@ "id": "021231", "content": "已知双曲线$\\dfrac{x^2}{64}-\\dfrac{y^2}{36}=1$的左、右焦点分别为$F_1$、$F_2$, 直线$l$过点$F_1$, 交双曲线的左支于$A$、$B$两点, 且$|AB|=m$, 求$\\triangle ABF_2$的周长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338234,7 +339952,9 @@ "id": "021232", "content": "设$P(x_0, y_0)$是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)右支上一点, $F_1$、$F_2$分别是左、右两个焦点, 求$|PF_1|$和$|PF_2|$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338253,7 +339973,9 @@ "id": "021233", "content": "设双曲线$\\Gamma$的两个焦点为$F_1(-\\sqrt 5, 0)$、$F_2(\\sqrt 5, 0)$, 点$P$是双曲线$\\Gamma$上的一点, 且满足$\\angle F_1PF_2=\\dfrac{\\pi}2$, $S_{\\triangle F_1PF_2}=4$, 求双曲线$\\Gamma$的标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338272,7 +339994,9 @@ "id": "021234", "content": "如果中心在原点, 对称轴在坐标轴上的等轴双曲线的一个焦点为$F_1(0,-6)$, 那么此双曲线的标准方程是\\blank{50}, 离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338291,7 +340015,9 @@ "id": "021235", "content": "双曲线$2 x^2-y^2=8$的焦点坐标是\\blank{50}, 两条渐近线的夹角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338310,7 +340036,9 @@ "id": "021236", "content": "若双曲线的中心在坐标原点, 它的一个焦点的坐标是$(-5,0)$, 两个顶点间的距离为$6$, 则此双曲线的方程是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}{36}-\\dfrac{y^2}{11}=1$}{$\\dfrac{x^2}{16}-\\dfrac{y^2}9=1$}{$\\dfrac{x^2}{11}-\\dfrac{y^2}{36}=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338329,7 +340057,9 @@ "id": "021237", "content": "在下列双曲线中, 以$y=\\pm \\dfrac 12 x$为渐近线的是\\bracket{20}.\n\\fourch{$\\dfrac{x^2}{16}-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4-\\dfrac{y^2}{16}=1$}{$\\dfrac{x^2}2-y^2=1$}{$x^2-\\dfrac{y^2}2=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338348,7 +340078,9 @@ "id": "021238", "content": "若方程$4 x^2+k y^2=4 k$表示双曲线, 则此双曲线的虚轴长等于\\bracket{20}.\n\\fourch{$2 \\sqrt k$}{$2 \\sqrt {-k}$}{$\\sqrt k$}{$\\sqrt {-k}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338367,7 +340099,9 @@ "id": "021239", "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0, b>0)$的两焦点为$F_1$、$F_2$, 若弦$AB$经过点$F_1$, 且$A$、$B$均在此双曲线的左支上, $|AB|=l$, 则$\\triangle ABF_2$的周长为\\bracket{20}.\n\\fourch{$4 a+l$}{$4 a+2 l$}{$2 a+l$}{$4 a-l$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338386,7 +340120,9 @@ "id": "021240", "content": "求以椭圆$\\dfrac{x^2}8+\\dfrac{y^2}5=1$的焦点为顶点, 以椭圆的顶点为焦点的双曲线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338405,7 +340141,9 @@ "id": "021241", "content": "已知双曲线的虚轴的长为$6$, 一条渐近线的方程为$3 x-y=0$, 求此双曲线的标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338424,7 +340162,9 @@ "id": "021242", "content": "求与双曲线$x^2-\\dfrac{y^2}4=1$有共同渐近线, 且过点$M(2,2)$的双曲线的标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338443,7 +340183,9 @@ "id": "021243", "content": "已知双曲线$\\dfrac{x^2}8-\\dfrac{y^2}{b^2}=1$的右焦点为点$F$, 若直线$x-y-3=0$经过点$F$, 求此双曲线渐近线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338462,7 +340204,9 @@ "id": "021244", "content": "已知双曲线$\\dfrac{x^2}9-\\dfrac{y^2}{16}=1$的两个焦点分别为$F_1$、$F_2$, 点$P$为此双曲线上一点, $|PF_1|\\cdot|PF_2|=32$, 求证: $PF_1 \\perp PF_2$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338481,7 +340225,9 @@ "id": "021245", "content": "已知双曲线$C$的中心在坐标原点, 若它的一个焦点为$(3,0)$, 一条渐近线的方程为$2 x-3 y=0$, 则此双曲线的方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338500,7 +340246,9 @@ "id": "021246", "content": "若双曲线的两渐近线的夹角为$\\dfrac{\\pi}3$, 则焦距与实轴长之比$\\dfrac ca=$\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338519,7 +340267,9 @@ "id": "021247", "content": "已知$F_1(-3,0)$、$F_2(3,0)$两点, 满足条件$|MF_1|+|MF_2|=2 m+1$的动点$M$的轨迹是椭圆, 满足条件$|NF_1|-|NF_2|=2 m-1$的动点$N$的轨迹是双曲线, 则实数$m$的取值范围是\\blank{50}.\n(4) 若双曲线$\\dfrac{x^2}4-\\dfrac{y^2}m=1$的渐近线方程为$y=\\pm \\dfrac{\\sqrt 3}2 x$, 则此双曲线的焦点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338538,7 +340288,9 @@ "id": "021248", "content": "若$a b \\neq 0$且$a \\neq b$, 则$a x-y+b=0$和$b x^2+a y^2=a b$所表示的曲线只可能是下图中的\\bracket{20}.\n\\fourch{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (0,0) ellipse (1 and 2);\n\\draw (-2.5,3) -- (1.5,-3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot (\\x,{sqrt(1+\\x*\\x/2)});\n\\draw [domain = -3:3] plot (\\x,{-sqrt(1+\\x*\\x/2)});\n\\draw (-3,-0.5) -- (0.5,3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (0,0) ellipse (2 and 1);\n\\draw (-1.5,3) -- (2.5,-3);\n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex,scale = 0.6]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,3) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -3:3] plot ({sqrt(0.5+0.25*\\x*\\x)},\\x);\n\\draw [domain = -3:3] plot ({-sqrt(0.5+0.25*\\x*\\x)},\\x);\n\\draw (-2,-3) -- (3,-0.5);\n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338557,7 +340309,9 @@ "id": "021249", "content": "已知曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0, b>0)$的两条渐近线均和圆$C: x^2+y^2-6 x+5=0$相切, 且双曲线的右焦点为圆$C$的圆心, 则该双曲线的方程为\\bracket{20}.\n\\fourch{$\\dfrac{x^2}5-\\dfrac{y^2}4=1$}{$\\dfrac{x^2}4-\\dfrac{y^2}5=1$}{$\\dfrac{x^2}3-\\dfrac{y^2}6=1$}{$\\dfrac{x^2}6-\\dfrac{y^2}3=1$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338576,7 +340330,9 @@ "id": "021250", "content": "已知$F_1(-8,3)$、$F_2(2,3)$两点, 动点$P$满足$|PF_1|-|PF_2|=2 a$. 当$a$分别为$3$和$5$时, 点$P$的轨迹分别为\\bracket{20}.\n\\twoch{双曲线和一条直线}{双曲线和一条射线}{双曲线的一支和一条直线}{双曲线的一支和一条射线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338595,7 +340351,9 @@ "id": "021251", "content": "从双曲线上任意一点$M$引实轴的平行线, 交它的渐近线于$P$、$Q$两点. 求证: $|MP|\\cdot|MQ|$是定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338614,7 +340372,9 @@ "id": "021252", "content": "已知点$M(x_0, y_0)$是双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$上任意一点, 直线$l$的方程是$x=\\dfrac{a^2}c$, 其中$c=\\sqrt {a^2+b^2}$. 求点$M$到焦点$F(c, 0)$的距离与它到直线$l$的距离之比.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338633,7 +340393,9 @@ "id": "021253", "content": "已知$F_1$、$F_2$为双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1(a>0, b>0)$的左、右焦点, 过$F_2$作垂直于$x$轴的直线, 交此双曲线于点$P$, 且$\\angle PF_1F_2=30^{\\circ}$, 求此双曲线的渐近线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338652,7 +340414,9 @@ "id": "021254", "content": "已知双曲线$\\dfrac{x^2}2-\\dfrac{y^2}{b^2}=1$($b>0$)的左、右焦点分别为$F_1$、$F_2$, 其一条渐近线方程为$y=x$. 若点$P(\\sqrt 3, y_0)$在该双曲线上, 则$\\overrightarrow{PF} \\cdot \\overrightarrow{PF_2}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338671,7 +340435,9 @@ "id": "021255", "content": "若双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}2=1(a>2)$的两条渐近线的夹角为$\\dfrac{\\pi}3$, 则此双曲线的离心率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338690,7 +340456,9 @@ "id": "021256", "content": "已知双曲线$\\dfrac{x^2}{12}-\\dfrac{y^2}4=1$的左、右焦点分别为$F_1$、$F_2$, 若点$M$在双曲线的左支上, 且$MF_1 \\perp x$轴, 则$F_1$到直线$MF_2$的距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338709,7 +340477,9 @@ "id": "021257", "content": "已知双曲线方程为$x^2-\\dfrac{y^2}4=1$, 过$P(1,0)$的直线$L$与双曲线只有一个公共点, 则$L$的条数共\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338728,7 +340498,9 @@ "id": "021258", "content": "求与双曲线$x^2-4 y^2=4$有相同的渐近线且焦距为$6$的双曲线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338747,7 +340519,9 @@ "id": "021259", "content": "若直线$y=k x+2$与双曲线$x^2-y^2=6$的右支交于不同的两点, 求实数$k$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338766,7 +340540,9 @@ "id": "021260", "content": "已知二次曲线$C_k$的方程:$\\dfrac{x^2}{9-k}+\\dfrac{y^2}{4-k}=1$.\\\\\n(1) 分别求出方程表示椭圆和双曲线的条件;\\\\\n(2) 若双曲线$C_k$与直线$y=x+1$有公共点且实轴最长, 求双曲线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338785,7 +340561,9 @@ "id": "021261", "content": "已知直线$l: y=a x+1$与双曲线$C: 3 x^2-y^2=1$相交于$A$、$B$两点.\\\\\n(1) 求实数$a$的取值范围;\\\\\n(2) 求当实数$a$为何值时, 以线段$AB$为直径的圆经过坐标原点.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338804,7 +340582,9 @@ "id": "021262", "content": "若$\\lg y$是$\\lg (x-3)$和$\\lg (x+3)$的平均数, 则点$(x, y)$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338823,7 +340603,9 @@ "id": "021263", "content": "已知$\\triangle ABC$中的两个顶点是$B(0,6)$、$C(0,-6), AB$边与$AC$边所在直线的斜率之积是$\\dfrac 49$, 求顶点$A$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338842,7 +340624,9 @@ "id": "021264", "content": "已知定点$A(3,0)$和定圆$B:(x+3)^2+y^2=16$, 动圆$C$与圆$B$外切, 且过点$A$, 求动圆的圆心$C$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338861,7 +340645,9 @@ "id": "021265", "content": "已知在$\\triangle ABC$中, $A$为动点, $B$、$C$两定点的坐标分别为$(-2,0)$、$(2,0)$, 且满足$\\sin C-\\sin B=\\dfrac 12 \\sin A$, 求动点$A$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338880,7 +340666,9 @@ "id": "021266", "content": "已知双曲线方程为$3 x^2-y^2=3$. 经过点$A(1,1)$能否作一条直线$l$, 使$l$与已知曲线交于$P$、$Q$两点, 且点$A$为线段$PQ$中点? 若存在, 写出$l$的方程; 若不存在, 说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338899,7 +340687,9 @@ "id": "021267", "content": "设双曲线$C:\\dfrac{x^2}{a^2}-y^2=1$($a>0$)与直线$l: x+y=1$相交于两个不同的点$A$、$B$.\\\\\n(1) 求双曲线$C$的离心率的取值范围;\\\\\n(2) 设直线$l$与$y$轴的交点为$P$, 且$\\overrightarrow{PA}=\\dfrac 5{12} \\overrightarrow{PB}$, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -338918,7 +340708,9 @@ "id": "021268", "content": "填写下表:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline 标准方程 & 图形 & 顶点 & 对称轴 & 焦点 & 准线 \\\\\n\\hline $y^2=2 p x$($p>0$) & \\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [right] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (-0.5,-2) -- (-0.5,2);\n\\draw [domain = -2:2] plot ({\\x*\\x/2},\\x);\n\\end{tikzpicture} & \\blank{40} & \\blank{40} & \\blank{40} & \\blank{40} \\\\\n\\hline $y^2=-2 p x$($p>0$)& \\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [right] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture} & \\blank{40} & \\blank{40} & \\blank{40} & \\blank{40} \\\\\n\\hline $x^2=2 p y$($p>0$)& \\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [right] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture}& \\blank{40} & \\blank{40} & \\blank{40} & \\blank{40} \\\\\n\\hline $x^2=-2 p y$($p>0$)&\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [right] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\end{tikzpicture} & \\blank{40} & \\blank{40} & \\blank{40} & \\blank{40} \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338937,7 +340729,9 @@ "id": "021269", "content": "到点$A(1,1)$和直线$l: x+2 y-3=0$距离相等的点的轨迹是\\bracket{20}.\n\\fourch{直线}{抛物线}{椭圆}{双曲线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -338956,7 +340750,9 @@ "id": "021270", "content": "抛物线$x^2=-32 y$的焦点坐标是\\blank{50}, 准线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338975,7 +340771,9 @@ "id": "021271", "content": "抛物线$y=4 x^2$的焦点坐标是\\blank{50}, 准线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -338994,7 +340792,9 @@ "id": "021272", "content": "抛物线$3 x^2+2 y=0$的焦点坐标是\\blank{50}, 准线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339013,7 +340813,9 @@ "id": "021273", "content": "根据下列条件, 写出顶点在原点, 焦点在坐标轴上的抛物线的标准方程:\\\\\n(1) 准线方程是$x=\\dfrac 14$;\\\\\n(2) 焦点到准线的距离是$2$;\\\\\n(3) 过点$(-3,4)$;\\\\\n(4) 过焦点且与$x$轴垂直的弦长是$16$;\\\\\n(5) 焦点在直线$3 x-4 y-12=0$上的抛物线标准方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339032,7 +340834,9 @@ "id": "021274", "content": "抛物线$y=a x^2$($a>0$)的焦点到准线的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339051,7 +340855,9 @@ "id": "021275", "content": "抛物线$y^2=m x$($m<0$)的焦点坐标是\\blank{50}, 准线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339070,7 +340876,9 @@ "id": "021276", "content": "设抛物线$y^2=2 p x$过点$M(2,2)$, 则点$M$到抛物线准线的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339089,7 +340897,9 @@ "id": "021277", "content": "若抛物线$y^2=2 p x$($p>0$)的焦点与椭圆$\\dfrac{x^2}6+\\dfrac{y^2}2=1$的右焦点重合, 则$p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339108,7 +340918,9 @@ "id": "021278", "content": "已知$O$为坐标原点, $F$为抛物线$y^2=4 x$的焦点, $A$为此抛物线上的一点, 若$\\overrightarrow{OA} \\cdot \\overline{AF}=-4$, 则点$A$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339127,7 +340939,9 @@ "id": "021279", "content": "若抛物线$y^2=4 x$上的点$P$到焦点$F$的距离为$4$, 则点$P$的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339146,7 +340960,9 @@ "id": "021280", "content": "已知点$F(\\dfrac 14, 0)$, 直线$l: x=-\\dfrac 14$, 点$B$是$l$上的动点, 若过$B$且垂直于$y$轴的直线与线段$BF$的垂直平分线交于点$M$, 则$M$点的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339165,7 +340981,9 @@ "id": "021281", "content": "方程$x^2 \\sin \\alpha+y^2 \\cos \\alpha=1$表示的曲线不可能是\\bracket{20}.\n\\fourch{直线}{抛物线}{圆}{双曲线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -339184,7 +341002,9 @@ "id": "021282", "content": "抛物线$y=a x^2$的准线方程是$y=2$, 则$a$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339203,7 +341023,9 @@ "id": "021283", "content": "已知抛物线$y^2=2 p x$的点$M$的横坐标为$2$. 若点$M$到焦点$F$的距离为 $5$, 则$p=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339222,7 +341044,9 @@ "id": "021284", "content": "已知抛物线$y^2=8 x$的焦点为$F$, $P$在此抛物线上, 且$|PF|=5$, 则点$P$的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339241,7 +341065,9 @@ "id": "021285", "content": "已知直线$l$垂直于$x$轴, 且交抛物线$y^2=4 x$于点$A$、$B$, 且$|AB|=4 \\sqrt 3$, 则直线$AB$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339260,7 +341086,9 @@ "id": "021286", "content": "点$P$到点$F(2,0)$的距离比它到直线$x+4=0$的距离小$2$, 则点$P$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339279,7 +341107,9 @@ "id": "021287", "content": "已知探照灯反射镜的轴截面是抛物线的一部分. 若某灯的灯口直径是$60 \\text{cm}$, 灯深是$40 \\text{cm}$, 则相应的抛物线的焦点到顶点的距离是\\blank{50}$\\text{cm}$.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339298,7 +341128,9 @@ "id": "021288", "content": "已知一隧道的顶部是抛物拱形, 拱高是$1$米, 跨度为$2$米, 建立适当的直角坐标系, 求相应坐标系下此拱形的抛物线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339317,7 +341149,9 @@ "id": "021289", "content": "已知正三角形$ABC$的顶点$A$位于坐标原点, 顶点$B$与$C$均在抛物线$y^2=2 x$上, 求$\\triangle ABC$的边长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339336,7 +341170,9 @@ "id": "021290", "content": "在抛物线$x^2=\\dfrac 14 y$上求一点$M$, 使点$M$到直线$y=4 x-5$的距离最短.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339355,7 +341191,9 @@ "id": "021291", "content": "已知抛物线$C$的顶点是原点, 它的焦点在$x$轴的正半轴上, 且拋物线$C$上动点$M$到点$A(\\dfrac 72, 4)$的距离与点$M$到焦点$F$的距离之和的最小值为$5$, 求抛物线$C$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339374,7 +341212,9 @@ "id": "021292", "content": "``直线与拋物线有且仅有一个公共点''是``直线和抛物线相切''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -339393,7 +341233,9 @@ "id": "021293", "content": "过定点$P(0,2)$, 作直线$l$与曲线$y^2=4 x$有且仅有$1$个公共点, 则这样的直线$l$共有\\blank{50}条.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339412,7 +341254,9 @@ "id": "021294", "content": "若直线$x-y-2=0$与抛物线$y^2=4 x$交于$A$、$B$两点, 则线段$AB$中点的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339431,7 +341275,9 @@ "id": "021295", "content": "设抛物线$y^2=4 x$截直线$y=2 x+m$所得的弦长为$3 \\sqrt 5$, 则实数$m=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339450,7 +341296,9 @@ "id": "021296", "content": "过点$Q(4,1)$作抛物线$y^2=8 x$的弦$AB, AB$恰好被点$Q$平分, 求$AB$所在直线的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339469,7 +341317,9 @@ "id": "021297", "content": "过点$Q(4,1)$作抛物线$y^2=8 x$的弦$AB$, 求弦$AB$的中点的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339488,7 +341338,9 @@ "id": "021298", "content": "求抛物线$y=2 x^2+1$上的动点$M$与点$N(0,-1)$所连线段的中点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339507,7 +341359,9 @@ "id": "021299", "content": "已知直线$y=x-2$与抛物线$y^2=a x$相交于$A$、$B$两点, 且$OA \\perp OB$, 求实数$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339526,7 +341380,9 @@ "id": "021300", "content": "直线$y=k x-2$交拋物线$y^2=8 x$于$A$、$B$两点, 若$AB$中点横坐标为 $2$, 求弦$|AB|$的长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339545,7 +341401,9 @@ "id": "021301", "content": "已知点$A$的坐标为$(0,4)$, 抛物线$C: y^2=2 p x$过点$M(1,2)$.\\\\\n(1) 若直线$AM$交抛物线$C$于另一点$N$, 求线段$MN$的长;\\\\\n(2) 直线$l$过$A$, 且与抛物线$C$只有一个公共点, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339564,7 +341422,9 @@ "id": "021302", "content": "已知圆$x^2+y^2+6 x-7=0$与抛物线$y^2=2 a x$的准线相切, 则实数$a=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339583,7 +341443,9 @@ "id": "021303", "content": "已知抛物线的顶点是双曲线$16 x^2-9 y^2=144$的中心, 它的焦点是双曲线的左顶点, 则此抛物线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339602,7 +341464,9 @@ "id": "021304", "content": "若过抛物线$y^2=-x$的焦点$F$的直线交拋物线于$A$、$B$两点, 且$A$、$B$在直线$x=\\dfrac 14$上的射影分别为$M$、$N$, 则$\\angle MFN=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339621,7 +341485,9 @@ "id": "021305", "content": "抛物线的顶点在原点, 以$x$轴为对称轴, 经过焦点且倾斜角为$135^{\\circ}$的直线, 被抛物线所截得的弦长为$8$, 则抛物线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339640,7 +341506,9 @@ "id": "021306", "content": "$P$为抛物线$y^2=2px$($p>0$)上任意一点, $F$为焦点, 则以$|PF|$为直径的圆与$y$轴的位置关系是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339659,7 +341527,9 @@ "id": "021307", "content": "过抛物线$y^2=4 x$的焦点作直线交抛物线于$A(x_1, y_1)$、$B(x_2, y_2)$两点, 且$x_1+x_2=6$, 则$|AB|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339678,7 +341548,9 @@ "id": "021308", "content": "若$AB$是抛物线$y^2=18 x$的一条过焦点$F$的弦, $|AB|=20$, $AD$、$BC$垂直于$y$轴, $D$、$C$分别为垂足, 则梯形$ABCD$的中位线的长是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339697,7 +341569,9 @@ "id": "021309", "content": "过抛物线$y^2=4 x$焦点的弦长为$6$, 求此弦中点的横坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339716,7 +341590,9 @@ "id": "021310", "content": "抛物线$y^2=8 x$的动弦$AB$的长为$16$, 求弦$AB$的中点$M$到$y$轴的最短距离.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339735,7 +341611,9 @@ "id": "021311", "content": "若经过抛物线$y^2=4 x$的焦点的弦被焦点分成长度为$m$和$n$的两部分, 求证$\\dfrac 1m+\\dfrac 1n$为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339754,7 +341632,9 @@ "id": "021312", "content": "过抛物线$x^2=2 p y(p>0)$的焦点的直线与抛物线交于$A, B$两点, 设$A(x_1, y_1)$, $B(x_2, y_2)$, 求证:$x_1 x_2$以及$y_1 y_2$都为定值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339773,7 +341653,9 @@ "id": "021313", "content": "抛物线$y^2=-2 p x(p>0)$上有一点$M$的横坐标是$-9$, 它到焦点的距离为 $10$, 则拋物线方程为\\blank{50}, $M$点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339792,7 +341674,9 @@ "id": "021314", "content": "过抛物线$y^2=-8 x$内一点$P(-1,1)$, 且被点$P$平分的抛物线的弦所在直线的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339811,7 +341695,9 @@ "id": "021315", "content": "直线$y=k x-2$交抛物线$y^2=8 x$于$A$、$B$两点, 若$AB$中点横坐标为 $2$, 则实数$k=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339830,7 +341716,9 @@ "id": "021316", "content": "若$AB$是抛物线$y=x^2$的一条过焦点的弦, 且$|AB|=4$, 则$AB$的中点到直线$y+1=0$的距离是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339849,7 +341737,9 @@ "id": "021317", "content": "抛物线$y^2=2 x$上到直线$x-y+3=0$距离最短的点的坐标为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339868,7 +341758,9 @@ "id": "021318", "content": "设直线$l$过点$A(1,2)$且与拋物线$y^2=4 x$只有一个公共点, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339887,7 +341779,9 @@ "id": "021319", "content": "若抛物线$x^2=4 y$的焦点弦长为$\\dfrac{16}3$, 求该焦点弦所在直线方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339906,7 +341800,9 @@ "id": "021320", "content": "若抛物线$y=a x^2-1$上有关于直线$y=-x$对称的不同两点, 求实数$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339925,7 +341821,9 @@ "id": "021321", "content": "设$P$、$Q$是抛物线$y^2=2 x$上满足$OP \\perp OQ$的任意两点, 其中$O$是坐标原点.\\\\\n(1) 求证: $PQ$所在直线与$x$轴交于定点;\\\\\n(2) 求$\\triangle POQ$面积的最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339944,7 +341842,9 @@ "id": "021322", "content": "在平面直角坐标系$x O y$中, 直线$l$与抛物线$y^2=2 x$相交于$A$、$B$两点.\\\\\n(1) 求证: ``若直线$l$经过点$T(3,0)$, 则$\\overrightarrow{OA} \\cdot \\overrightarrow{OB}=3$''是真命题;\\\\\n(2) 写出第(1)题中命题的逆命题, 判断它是真命题还是假命题, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -339963,7 +341863,9 @@ "id": "021323", "content": "若动点$M$到点$F(5,0)$的距离与动点$M$到直线$x=-3$的距离之差等于$2$, 则点$M$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -339982,7 +341884,9 @@ "id": "021324", "content": "过抛物线$y^2=2 p x(p>0)$的焦点$F$的直线$l$交抛物线于相异两点$A(x_1, y_1)$、$B(x_2, y_2)$, 则$y_1 y_2=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340001,7 +341905,9 @@ "id": "021325", "content": "抛物线$y^2=4 x$关于直线$x+y=0$的对称曲线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340020,7 +341926,9 @@ "id": "021326", "content": "设抛物线$y^2=8 x$的焦点为$F$, 准线为$l$, $P$为抛物线上一点, $PA \\perp l, A$为垂足. 如果直线$AF$的斜率为$-\\sqrt 3$, 则$|PF|=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340039,7 +341947,9 @@ "id": "021327", "content": "设拋物线$y^2=8 x$的准线与$x$轴交于点$Q$, 若经过点$Q$的直线$l$与此拋物线有公共点, 则直线$l$的斜率取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340058,7 +341968,9 @@ "id": "021328", "content": "如图, 一位运动员在距篮下$4$米处跳起投篮, 球运行的路线是抛物线, 当球运行的水平距离为$2.5$米时, 达到最大高度为$3.5$米, 然后准确落入篮框, 已知篮框中心到地面的距离为$3.05$米.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-3,0) -- (3,0) node [below] {$x$};\n\\draw [->] (0,-0.6) -- (0,4) node [left] {$y$};\n\\draw (0,0) node [below right] {$O$};\n\\draw (-2.5,0) -- (-2.5,-0.9);\n\\draw [<->] (-2.5,-0.3) -- (0,-0.3) node [midway, fill = white] {$2.5$米};\n\\draw [<->] (-2.5,-0.7) -- (1.5,-0.7) node [midway, fill = white] {$4$米};\n\\draw [dashed] (1.5,-0.9) -- (1.5,3.05);\n\\draw (1.5,3.05) ellipse (0.2 and 0.05) (1.7,3.05) -- (1.9,3.05) (1.9,2.95) --++ (0,0.6);\n\\draw (1.95,3.05) --++ (0.4,0);\n\\draw [<->] (2.15,3.05) -- (2.15,0) node [midway, fill = white, rotate = 90] {$3.05$米};\n\\draw [domain = -2.5:1.5,dashed] plot (\\x,{3.5-0.2*\\x*\\x});\n\\filldraw [fill = white, draw = black] (-0.5,{3.5-0.2*0.25}) circle (0.1);\n\\end{tikzpicture}\n\\end{center}\n(1) 建立如图所示的平面直角坐标系, 求抛物线的方程;\\\\\n(2) 该运动员身高为$1.8$米, 在这次跳投中, 球在头顶上方$0.25$米处出手, 问球出手时, 他跳离地面的高度是多少?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340077,7 +341989,9 @@ "id": "021329", "content": "已知$A$、$B$、$C$三点在曲线$y=\\sqrt x$上, 其横坐标依次为$1$、$m$、\n$4$($10$)的焦点为$F$, $A$是抛物线上横坐标为$4$、且位于$x$轴上方的点, $A$到抛物线准线的距离等于$5$, 过$A$作$AB$垂直于$y$轴, 垂足为$B$, $OB$的中点为$M$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (-1,0) -- (5,0) node [below] {$x$};\n\\draw [->] (0,-5) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw (4,4) node [above] {$A$} coordinate (A);\n\\draw (1,0) node [below] {$F$} coordinate (F);\n\\draw (0,4) node [left] {$B$} coordinate (B);\n\\draw ($(O)!0.5!(B)$) node [left] {$M$} coordinate (M);\n\\draw ($(A)!(M)!(F)$) node [below right] {$N$} coordinate (N);\n\\draw (A) -- (B) (A) -- (F) (M) -- (N);\n\\draw [domain = -4.3:4.3] plot ({\\x*\\x/4},\\x);\n\\end{tikzpicture}\n\\end{center}\n(1) 求抛物线方程;\\\\\n(2) 过$M$作$MN \\perp FA$, 垂足为$N$, 求点$N$的坐标;\\\\\n(3) 以$M$为圆心, $MB$为半径作圆$M$. 当$K(m, 0)$是$x$轴上一动点时, 讨论直线$AK$与圆$M$的位置关系.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340115,7 +342031,9 @@ "id": "021331", "content": "下列各组方程中表示相同曲线的是\\bracket{20}.\n\\fourch{$y=x$, $\\dfrac yx=1$}{$y=x$, $y=\\sqrt {x^2}$}{$|x|=|y|$, $\\sqrt x=\\sqrt y$}{$|x|=|y|$, $x^2=y^2$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -340134,7 +342052,9 @@ "id": "021332", "content": "方程$y=-\\sqrt {x^2-2 x+1}$的图形是下图中的\n\\fourch{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (-0.1,1) -- (0,1) node [right] {$1$};\n\\draw (-0.1,-1) -- (0,-1) node [right] {$-1$};\n\\draw (-1,2) -- (2,-1) (-1,-2) -- (2,1); \n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (-0.1,1) -- (0,1) node [right] {$1$};\n\\draw (-0.1,-1) -- (0,-1) node [right] {$-1$};\n\\draw (1,0) -- (2,1) (1,0) -- (2,-1); \n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (-0.1,1) -- (0,1) node [right] {$1$};\n\\draw (-0.1,-1) -- (0,-1) node [right] {$-1$};\n\\draw (-1,-2) -- (1,0) (1,0) -- (2,-1); \n\\end{tikzpicture}}{\\begin{tikzpicture}[>=latex, scale = 0.7]\n\\draw [->] (-2,0) -- (2,0) node [below] {$x$};\n\\draw [->] (0,-2) -- (0,2) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw (1,0.1) -- (1,0) node [below] {$1$};\n\\draw (-1,0.1) -- (-1,0) node [below] {$-1$};\n\\draw (-0.1,1) -- (0,1) node [right] {$1$};\n\\draw (-0.1,-1) -- (0,-1) node [right] {$-1$};\n\\draw (-1,2) -- (1,0) (-1,-2) -- (1,0); \n\\end{tikzpicture}}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340153,7 +342073,9 @@ "id": "021333", "content": "``点$M(a, b)$在曲线$y^2=x$上''是``点$M(a, b)$在曲线$y=\\sqrt x$上''的\\bracket{20}.\n\\twoch{充分非必要条件}{必要非充分条件}{充要条件}{既非充分又非必要条件}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -340172,7 +342094,9 @@ "id": "021334", "content": "已知$A(2,1)$、$B(1,-1)$, 则射线$AB$的方程是\\blank{50}, 线段$AB$的方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340191,7 +342115,9 @@ "id": "021335", "content": "曲线$y=2 x^2$关于点$(1,2)$对称的曲线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340210,7 +342136,9 @@ "id": "021336", "content": "已知动点$P$到点$F_1(4,0)$的距离与它到$y$轴的距离相等, 则点$P$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340229,7 +342157,9 @@ "id": "021337", "content": "已知$A$、$B$两点的坐标是$(1,0)$、$(-1,0)$, 动点$M$满足$\\overrightarrow{MA} \\perp \\overrightarrow{MB}$, 则动点$M$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340248,7 +342178,9 @@ "id": "021338", "content": "已知等腰三角形底边的两个端点的坐标分别是$B(4,2)$、$C(-2,0)$, 则第三个顶点$A$的轨迹方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340267,7 +342199,9 @@ "id": "021339", "content": "已知过抛物线$y^2=4 x$的焦点$F$的直线交抛物线于$A$、$B$两点, 过原点$O$作$\\overrightarrow{OM}$, 使$\\overrightarrow{OM} \\perp \\overrightarrow{AB}$, 垂足为$M$, 求点$M$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340286,7 +342220,9 @@ "id": "021340", "content": "若$\\triangle ABC$的两个顶点$A(-3,0)$、$B(6,0)$, 点$C$在曲线$x^2+(y-6)^2=1$上运动. 求$\\triangle ABC$的重心的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340305,7 +342241,9 @@ "id": "021341", "content": "已知$A(3,1)$、$B(-1,3)$两点, $O$为坐标原点, 动点$C$满足$\\overrightarrow{OC}=\\alpha \\overrightarrow{OA}+\\beta \\overrightarrow{OB}$, 其中$\\alpha$、$\\beta \\in \\mathbf{R}$, 且$\\alpha+\\beta=1$, 求点$C$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340324,7 +342262,9 @@ "id": "021342", "content": "长为$4$的线段$AB$的两端点分别在$x$轴正半轴、$y$轴正半轴上滑动, 求$AB$中点的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340343,7 +342283,9 @@ "id": "021343", "content": "若动点$P(x, y)$到定点$A(3,4)$的距离比$P$到$x$轴的距离多$1$, 求动点$P$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340362,7 +342304,9 @@ "id": "021344", "content": "已知曲线$\\begin{cases}x=3 \\cos \\theta,\\\\y=2 \\sin \\theta,\\end{cases}$($\\theta$为参数), 则$\\theta=\\dfrac{\\pi}6$时对应的点$P$的坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340381,7 +342325,9 @@ "id": "021345", "content": "参数方程$\\begin{cases}x=3 t^2+4, \\\\y=t^2-2,\\end{cases}$($0 \\leq t \\leq 3$)所表示的曲线是\\bracket{20}.\n\\fourch{双曲线的一支}{线段}{圆弧}{射线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -340400,7 +342346,9 @@ "id": "021346", "content": "参数方程$\\begin{cases}x=1+2 \\cos \\theta, \\\\y=2 \\sin \\theta,\\end{cases}$($\\theta$为参数)的普通方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340419,7 +342367,9 @@ "id": "021347", "content": "下列参数方程($t$为参数)中, 与$x^2-y=0$表示同一曲线的是\\bracket{20}.\n\\fourch{$\\begin{cases}x=t^2, \\\\y=t\\end{cases}$}{$\\begin{cases}x=\\sqrt {|t|}, \\\\y=t\\end{cases}$}{$\\begin{cases}x=\\sin t, \\\\y=\\sin ^2 t\\end{cases}$}{$\\begin{cases}x=\\tan t, \\\\y=\\dfrac{1-\\cos 2 t}{1+\\cos 2 t}\\end{cases}$}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -340438,7 +342388,9 @@ "id": "021348", "content": "参数方程$\\begin{cases}x=2 \\sqrt m,\\\\y=1+\\sqrt {1-m},\\end{cases}$($m$为参数)的普通方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340457,7 +342409,9 @@ "id": "021349", "content": "两曲线$\\begin{cases}x=\\cos ^2 \\theta,\\\\y=\\sin ^2 \\theta,\\end{cases}$($\\theta$为参数)和$\\begin{cases}x=2 \\cos t, \\\\y=2 \\sin t,\\end{cases}$($t$为参数)的交点个数是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340476,7 +342430,9 @@ "id": "021350", "content": "曲线$\\begin{cases}x=1+2 \\cos ^2 \\theta, \\\\y=\\sqrt 2 \\sin \\theta,\\end{cases}$($\\theta$为参数, $\\theta \\in \\mathbf{R}$)与直线$y=x$的交点坐标是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340495,7 +342451,9 @@ "id": "021351", "content": "若直线$\\begin{cases}x=4+a t, \\\\y=b t,\\end{cases}$($t$为参数)与圆$x^2+y^2-4 x+1=0$相切, 则直线的倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340514,7 +342472,9 @@ "id": "021352", "content": "拋物线$\\begin{cases}x=-4 t^2+1,\\\\y=4 t,\\end{cases}$($t$为参数)在$y$轴上截得的弦长等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340533,7 +342493,9 @@ "id": "021353", "content": "方程$x^2+y^2-4 t x-2 t y+5 t^2-4=0$($t$为参数)所表示的一族圆的圆心轨迹是\\bracket{20}.\n\\fourch{一个定点}{一个椭圆}{一条抛物线}{一条直线}", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "选择题", "ans": "", "solution": "", @@ -340552,7 +342514,9 @@ "id": "021354", "content": "已知曲线$C$的参数方程是$\\begin{cases}x=3 t, \\\\y=2 t^2+1,\\end{cases}$($t$为参数).\\\\\n(1) 判断点$M_1(0,1)$, $M_2(5,4)$与曲线$C$的位置关系;\\\\\n(2) 已知点$M_3(6, a)$在曲线$C$上, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340571,7 +342535,9 @@ "id": "021355", "content": "设$a$、$b$是非零常数, 由参数方程$\\begin{cases}x=\\dfrac a{\\cos \\alpha},\\\\y=b \\tan \\alpha,\\end{cases}$($\\alpha \\neq k \\pi+\\dfrac{\\pi}2$, $k \\in \\mathbf{Z}$)表示的轨迹在什么曲线上?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340590,7 +342556,9 @@ "id": "021356", "content": "将参数方程$\\begin{cases}x=2+\\cos \\theta, \\\\y=-\\cos \\theta,\n\\end{cases}$($\\theta$为参数)化为普通方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340609,7 +342577,9 @@ "id": "021357", "content": "若直线$l$的参数方程为$\\begin{cases}x=-t \\cos \\dfrac{\\pi}6+2, \\\\y=t \\sin \\dfrac{\\pi}6,\\end{cases}$($t$为参数), 则其倾斜角为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340628,7 +342598,9 @@ "id": "021358", "content": "已知$\\begin{cases}x=2+\\cos \\theta, \\\\y=\\sin \\theta,\\end{cases}$($\\theta$为参数), 则$\\sqrt {(x-5)^2+(y+4)^2}$的最大值是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340647,7 +342619,9 @@ "id": "021359", "content": "点$P(x, y)$在椭圆$\\dfrac{x^2}4+y^2=1$上, 则$2 x+y$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340666,7 +342640,9 @@ "id": "021360", "content": "以原点为圆心, $1$为半径作一个圆, 设定点$A$的坐标为$(2,0)$, $B$为圆上任意一点, $M$为线段$AB$的中点, 求点$M$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340685,7 +342661,9 @@ "id": "021361", "content": "动点$M$作匀速直线运动, 它在$x$轴和$y$轴方向的分速度分别为$9$和$12$, 运动开始时, 点$M$位于$A(1,1)$, 求点$M$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340704,7 +342682,9 @@ "id": "021362", "content": "设$P$是椭圆$\\dfrac{x^2}{36}+\\dfrac{y^2}4=1$在第一象限部分的弧$AB$上的一点, 求使四边形$OAPB$的面积最大的点$P$的坐标.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340723,7 +342703,9 @@ "id": "021363", "content": "$A$为椭圆$\\dfrac{x^2}{25}+\\dfrac{y^2}9=1$上任一点, $B$为圆$(x-1)^2+y^2=1$上任一点, 求$|AB|$的最大值与最小值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340742,7 +342724,9 @@ "id": "021364", "content": "抛物线$y^2=4 x$的内接三角形的一个顶点在原点, 其重心恰是拋物线的焦点, 求该内接三角形的周长.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340761,7 +342745,10 @@ "id": "021365", "content": "函数$y=\\dfrac 1x$从$x=1$到$x=2$的平均变化率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340780,7 +342767,10 @@ "id": "021366", "content": "函数$y=x^2-x$在区间$[-2, t]$上的平均变化率为$2$, 则$t=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340799,7 +342789,10 @@ "id": "021367", "content": "函数$y=x^2$在$x=1$处的导数为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340818,7 +342811,10 @@ "id": "021368", "content": "已知$f(x)=\\dfrac 2x$, 且$f'(m)=-\\dfrac 12$, 则实数$m$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340837,7 +342833,10 @@ "id": "021369", "content": "直线运动的物体, 从时间$t$到$t+\\Delta t$时, 物体的位移是$\\Delta s$, 那么$\\displaystyle \\lim _{\\Delta t \\to 0} \\dfrac{\\Delta s}{\\Delta t}$为\\bracket{20}.\n\\onech{从时间$t$到$t+\\Delta t$时, 物体的平均速度}{在$t$时刻该物体的瞬时速度}{当时间为$0$时物体的瞬时速度}{从时间$0$到$\\Delta t$时, 物体的平均速度}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "选择题", "ans": "", "solution": "", @@ -340856,7 +342855,10 @@ "id": "021370", "content": "已知函数$f(x)=3 x^2+5$.\\\\\n(1) 求$f(x)$从$0.1$到$0.2$的平均变化率;\\\\\n(2) 求$f(x)$在区间$[x_0, x_0+\\triangle x]$上的平均变化率.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340875,7 +342877,10 @@ "id": "021371", "content": "设$f(x)=10 x^3$, 试按定义求$f'(-1)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340894,7 +342899,10 @@ "id": "021372", "content": "自由落体运动中物体下落的距离$d$(单位: 米)与时间$t$(单位: 秒)近似满足函数关系$d=5 t^2$.\\\\\n(1) 求物体在$[3,5]$时间段内的平均速度;\\\\\n(2) 求物体在$t=4$时的瞬时速度;\\\\\n(3) 求物体在$t=a$时的瞬时速度.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340913,7 +342921,10 @@ "id": "021373", "content": "将石子投入水中, 水面产生的圆形波纹不断扩散.\\\\\n(1) 当半径$r$从$a$增加到$a+h$($h>0$)时, 求圆周长相对于半径的平均变化率;\\\\\n(2) 当半径$r=a$时, 求圆周长相对于半径的瞬时变化率.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340932,7 +342943,10 @@ "id": "021374", "content": "已知$f'(x_0)=2$, 求$\\displaystyle\\lim _{k \\to 0} \\dfrac{f(x_0-\\dfrac 12 k)-f(x_0)}k$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -340951,7 +342965,10 @@ "id": "021375", "content": "已知$f(x)=x^2+1$, 则曲线$y=f(x)$在$P(1,2)$处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340970,7 +342987,10 @@ "id": "021376", "content": "已知函数$y=f(x)$在$x=1$处的切线方程为$y=4 x-3$, 则$f(1)=$\\blank{50}, $f'(1)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -340989,7 +343009,10 @@ "id": "021377", "content": "函数$y=f(x)$的图像如图所示. \n\\begin{center}\n\\begin{tikzpicture}[>=latex,scale = 0.5]\n\\draw [->] (0,0) -- (6,0) node [below] {$x$};\n\\draw [->] (0,-0) -- (0,5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\foreach \\i in {1,2,...,5} {\\draw (\\i,0.2) -- (\\i,0);};\n\\foreach \\i in {1,2,...,4} {\\draw (0.2,\\i) -- (0,\\i);};\n\\draw (1,0) node [below] {$1$} (5,0) node [below] {$5$};\n\\draw (0,1) node [left] {$1$} (0,4) node [left] {$4$};\n\\draw [dashed] (1,0) -- (1,1) node [right] {$P$} coordinate (P) -- (0,1) (5,0) -- (5,4) node [above] {$Q$} coordinate (Q) -- (0,4);\n\\draw ($(P)!{-0.1}!(Q)$) -- ($(P)!1.1!(Q)$);\n\\draw [domain = 0.6:5.4] plot (\\x,{-1/8*(\\x-6)*(\\x-6)+33/8}) node [right] {$y=f(x)$};\n\\end{tikzpicture}\n\\end{center}\n(1) 求割线$PQ$的斜率;\\\\\n(2) 当点$Q$沿曲线向点$F$运动时, 割线$PQ$的斜率会变大还是变小?", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341008,7 +343031,10 @@ "id": "021378", "content": "借助函数图像, 判断下列导数的正负:\\\\\n(1) $f'(-\\dfrac{\\pi}4)$, 其中$f(x)=\\cos x$;\\\\\n(2) $f'(3)$, 其中$f(x)=\\ln x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341027,7 +343053,10 @@ "id": "021379", "content": "已知曲线$C: f(x)=x^3+x$.\\\\\n(1) 求曲线$C$在点$(1,2)$处的切线的方程;\\\\\n(2) 设曲线上任意一点处切线的倾斜角为$\\alpha$, 求$\\alpha$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341046,7 +343075,10 @@ "id": "021380", "content": "根据导数的几何意义, 求函数$y=\\sqrt {4-x^2}$在下列各点处的导数:\\\\\n(1) $x=-1$;\\\\\n(2) $x=0$;\\\\ \n(3) $x=1$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341065,7 +343097,10 @@ "id": "021381", "content": "过曲线$y=x^2$上某点$P$的切线满足下列条件, 分别求出$P$点.\\\\\n(1) 与$x$轴成$135^{\\circ}$的倾斜角;\\\\\n(2) 垂直于直线$2 x-6 y+5=0$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341084,7 +343119,10 @@ "id": "021382", "content": "已知曲线$y=x^4+a x^2+1$在点$(-1, a+2)$处的切线的斜率是$8$, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341103,7 +343141,10 @@ "id": "021383", "content": "求过点$(-1,0)$且与曲线$y=x^2+x+1$相切的直线方程.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341122,7 +343163,10 @@ "id": "021384", "content": "已知$f(x)=\\ln x$且$f'(x_0)=\\dfrac 1{x_0^2}$, 则$x_0=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341141,7 +343185,10 @@ "id": "021385", "content": "曲线$y=\\mathrm{e}^x$在$P(0,1)$处的切线方程是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341160,7 +343207,10 @@ "id": "021386", "content": "若点$P$是曲线$y=\\mathrm{e}^x$上任意一点, 则点$P$到直线$y=x$的最小距离为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341179,7 +343229,10 @@ "id": "021387", "content": "已知函数$f(x)=x^\\alpha$(其中$\\alpha$是常数), 若$f'(-1)=-4$, 则$\\alpha$的值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341198,7 +343251,10 @@ "id": "021388", "content": "设$f_0(x)=\\sin x$, $f_1(x)=f_0'(x)$, $f_2(x)=f_1'(x)$, $\\cdots$, $f_{n+1}(x)=f_n'(x)$, $n \\in \\mathbf{N}$, 则$f_{2022}(x)=$\\bracket{20}.\n\\fourch{$\\sin x$}{$-\\sin x$}{$\\cos x$}{$-\\cos x$}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "选择题", "ans": "", "solution": "", @@ -341217,7 +343273,10 @@ "id": "021389", "content": "求下列函数的导数:\\\\\n(1) $y=2022$;\\\\\n(2) $y=\\sin 60^{\\circ}$;\\\\\n(3) $y=x \\sqrt x$;\\\\\n(4) $y=\\dfrac 1{\\sqrt [3]{x^2}}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341236,7 +343295,10 @@ "id": "021390", "content": "求余弦函数$y=\\cos x$的所有驻点.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341255,7 +343317,10 @@ "id": "021391", "content": "求证: 函数$f(x)=\\ln x$与函数$f(x)=\\mathrm{e}^x$都没有驻点.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341274,7 +343339,10 @@ "id": "021392", "content": "已知两条曲线$y=\\sin x$与$y=\\cos x$, 是否存在这两条曲线的一个公共点, 使在这一点处, 两条曲线的切线互相垂直? 如果存在, 请求出满足题意的所有公共点; 如果不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341293,7 +343361,10 @@ "id": "021393", "content": "已知$f(x)=a x^3+9 x^2+6 x-7$, 若$f'(-1)=4$, 则实数$a$的值等于\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341312,7 +343383,10 @@ "id": "021394", "content": "曲线$y=\\dfrac{\\sin x}{\\sin x+\\cos x}-\\dfrac 12$在点$(\\dfrac{\\pi}4, 0)$处的切线的斜率为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341331,7 +343405,10 @@ "id": "021395", "content": "曲线$y=x \\ln x$上的点到直线$x-y-2=0$的距离的最小值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341350,7 +343427,10 @@ "id": "021396", "content": "已知函数$f(x)=f'(\\dfrac{\\pi}4) \\cos x+\\sin x$, 则$f(\\dfrac{\\pi}4)=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341369,7 +343449,10 @@ "id": "021397", "content": "设函数$f(x)=\\dfrac{\\sin \\theta}3 x^3+\\dfrac{\\sqrt 3 \\cos \\theta}2 x^2+\\tan \\theta$, 其中$\\theta \\in[0, \\dfrac{5 \\pi}{12}]$, 则导数$f'(1)$的取值范围是\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341388,7 +343471,10 @@ "id": "021398", "content": "求下列函数的导数:\\\\\n(1) $y=x^2+2^x$;\\\\\n(2) $y=\\sqrt x-\\ln x$;\\\\\n(3) $y=(x^2+1)(x-1)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341407,7 +343493,10 @@ "id": "021399", "content": "求下列函数的导数:\\\\\n(1) $y=x^2 \\mathrm{e}^x$;\\\\\n(2) $y=\\sqrt x \\sin x$;\\\\\n(3) $y=x \\ln x$;\\\\\n(4) $y=\\lg x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341426,7 +343515,10 @@ "id": "021400", "content": "求下列函数的导数:\\\\\n(1) $y=\\dfrac{\\sin x}x$;\\\\\n(2) $y=\\dfrac{x^2}{\\ln x}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341445,7 +343537,10 @@ "id": "021401", "content": "设$a, b$是实数. 已知函数$f(x)=a x^2+b x+3$的导函数为$f'(x)=2 x-8$.\\\\\n(1) 求$a, b$的值;\\\\\n(2) 设函数$g(x)=\\mathrm{e}^x \\sin x+f(x)$, 求曲线$g(x)$在$x=0$处的切线方程.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341464,7 +343559,10 @@ "id": "021402", "content": "分别在下列条件中, 求复合函数.\\\\\n(1) 已知函数$f(x)=2 x$, $g(x)=x^2$, 则$f(g(x))=$\\blank{50}, $g(f(x))=$\\blank{50};\\\\\n(2) 已知函数$f(x)=3^x$, $g(x)=\\sin x$, 则$f(g(x))=$\\blank{50}, $g(f(x))=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341483,7 +343581,10 @@ "id": "021403", "content": "曲线$y=\\ln (2-x)$在点$(1,0)$处的切线方程为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341502,7 +343603,10 @@ "id": "021404", "content": "求下列函数的导数:\\\\\n(1) $y=(5 x-3)^4$;\\\\\n(2) $y=\\dfrac 1{(1-3 x)^4}$;\\\\\n(3) $y=\\sqrt [4]{\\dfrac 1{3 x+1}}$;\\\\\n(4) $y=\\mathrm{e}^{-0.02 x+1}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341521,7 +343625,10 @@ "id": "021405", "content": "求下列函数的导数:\\\\\n(1) $y=\\ln \\sqrt {1+2 x}$;\\\\\n(2) $y=\\dfrac{x^2}{(2 x+1)^3}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341540,7 +343647,10 @@ "id": "021406", "content": "求下列函数的导数:\\\\\n(1) $y=\\mathrm{e}^{2 x} \\sin 3 x$;\\\\\n(2) $y=\\ln \\sqrt {\\dfrac{1+x}{1-x}}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341559,7 +343669,10 @@ "id": "021407", "content": "求下列函数的导数:\\\\\n(1) $y=\\cos ^2 x$;\\\\\n(2) $y=\\ln (x \\cdot 3^{x+1}+3^x)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341578,7 +343691,10 @@ "id": "021408", "content": "试用两种不同的方法求下列函数的导数:\\\\\n(1) $y=\\dfrac{2-x}{x+1}$;\\\\\n(2) $y=\\sin x \\cos x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341597,7 +343713,10 @@ "id": "021409", "content": "求下列曲线在点$P$处的切线方程:\\\\\n(1) $y=3^{1-2 x}$, $P(0,3)$;\\\\\n(2) $y=\\log_2(3 x-4)$, $P(2,1)$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341616,7 +343735,10 @@ "id": "021410", "content": "已知函数$y=f(x)$是偶函数, 且当$x<0$时, $f(x)=\\ln (-x)+3 x$, 求曲线$y=f(x)$在点$(1,-3)$处的切线方程.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341635,7 +343757,10 @@ "id": "021411", "content": "利用导数研究下列函数的单调性, 并说明所得结果与你之前的认识是否一致.\\\\\n(1) $y=2^x$;\\\\\n(2) $y=x+\\dfrac 1x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341654,7 +343779,10 @@ "id": "021412", "content": "利用导数研究下列函数的单调性.\\\\\n(1) $y=x^3-x$;\\\\\n(2) $y=x+\\dfrac 1{x^2}$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341673,7 +343801,10 @@ "id": "021413", "content": "研究下列函数的单调性.\\\\\n(1) $y=x+\\sin x$;\\\\\n(2) $y=2^x-x \\ln 4$;\\\\\n(3) $y=x \\ln x$.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341692,7 +343823,10 @@ "id": "021414", "content": "研究下列函数的单调性.\\\\\n(1) $y=\\dfrac{\\sin x}x$($00$).", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341711,7 +343845,10 @@ "id": "021415", "content": "已知函数$f(x)=\\dfrac{\\ln x+k}{\\mathrm{e}^x}$($k$为常数, $\\mathrm{e}$是自然对数的底数), 曲线$y=f(x)$在点$(1, f(1))$处的切线与$x$轴平行.\\\\\n(1) 求$k$的值;\\\\\n(2) 求$f(x)$的单调区间.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341730,7 +343867,10 @@ "id": "021416", "content": "设$a$是实数. 已知函数$f(x)=2 x^3+a x-1$.\\\\\n(1) 若函数$y=f(x)$在$\\mathbf{R}$上为严格增函数, 求$a$的取值范围;\\\\\n(2) 若函数$y=f(x)$在$(-1,1)$上为严格减函数, 求$a$的取值范围;\\\\\n(3) 若函数$y=f(x)$的单调减区间为$[-1,1]$, 求$a$的值;\\\\\n(4) 若函数$y=f(x)$在区间$(-1,1)$上不是单调函数, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341749,7 +343889,10 @@ "id": "021417", "content": "已知函数$f(x)=\\dfrac{\\ln x}x$, 实数$a, b$满足$a3$, 试比较$a^b$与$b^a$的大小, 并说明理由.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341768,7 +343911,10 @@ "id": "021418", "content": "设$a$是实数, 若$x=2$是函数$f(x)=x^3-3 a x+2$的极小值点, 那么$y=f(x)$的极大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341787,7 +343933,10 @@ "id": "021419", "content": "设$a$是实数, 若函数$f(x)=a \\mathrm{e}^x-\\sin x$在$x=\\pi$处有极值, 则$a$的值为\\blank{50}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341806,7 +343955,10 @@ "id": "021420", "content": "设$a$是实数, 若函数$f(x)=x^3-3 a x$在区间$(-1,2)$上仅有一个极值点, 则$a$的取值范围为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341825,7 +343977,10 @@ "id": "021421", "content": "三次函数$y=f(x)$的$x^3$的系数为$1$, 且函数的大致图像如图所示, 其中$-1,2$是两个零点, $x_1, x_2$分别是极大值与极小值点, 则$x_1^2+x_2^2=$\\blank{50}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex, yscale = 0.4]\n\\draw [->] (-1.2,0) -- (2.2,0) node [below] {$x$};\n\\draw [->] (0,-3) -- (0,1.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -1.2:2.2,samples = 100] plot (\\x,{\\x*(\\x+1)*(\\x-2)});\n\\draw (-1,0) node [above left] {$-1$} (2,0) node [above left] {$2$};\n\\draw [dashed] ({(1-sqrt(7))/3},{(7*sqrt(7)-10)*2/27}) -- ({(1-sqrt(7))/3},0) node [below] {$x_1$};\n\\draw [dashed] ({(1+sqrt(7))/3},{(-7*sqrt(7)-10)*2/27}) -- ({(1+sqrt(7))/3},0) node [above] {$x_2$};\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -341844,7 +343999,10 @@ "id": "021422", "content": "求函数$f(x)=x^3-3 x+2$的单调区间和极值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341863,7 +344021,10 @@ "id": "021423", "content": "求函数$f(x)=2 x+\\dfrac 8x$的单调区间和极值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -341882,7 +344043,10 @@ "id": "021424", "content": "设$a$是实数. 已知函数$f(x)=a x+\\cos x(-\\pi0, \\\\h(x), & x<0\\end{cases}$是奇函数, 则函数$y=h(x)$的最大值为\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -342015,7 +344197,10 @@ "id": "021431", "content": "已知定义域为$\\mathbf{R}$的函数$y=f(x)$的图像是一条连续不断的曲线, 给出下列命题:\\\\\n\\textcircled{1} 若$y=f(x)$存在驻点$x=x_0$, 则$x=x_0$一定是该函数的一个极值点;\\\\\n\\textcircled{2} 若$y=f(x)$在$\\mathbf{R}$上有且仅有一个极大值点$x=x_0$, 则$f(x_0)$就是该函数的最大值;\\\\\n\\textcircled{3} ``函数$y=f(x)$有极小值点''是``$y=f(x)$有最小值''的必要非充分条件;\\\\\n\\textcircled{4} 若$y=f(x)$是严格增函数, 则该函数没有驻点;\\\\\n\\textcircled{5} 若$y=f(x)$有极值点, 则该函数不是单调函数.\n则上述命题中, 正确的有\\blank{50}(填写所有正确命题的序号).", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "填空题", "ans": "", "solution": "", @@ -342034,7 +344219,10 @@ "id": "021432", "content": "设$a$是实数. 已知函数$f(x)=-x^2-2 x+3$在区间$[a, 2]$上的最大值为$\\dfrac{15}4$, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342053,7 +344241,10 @@ "id": "021433", "content": "设$a$是实数. 已知函数$f(x)=a x-\\ln x$, 当$x \\in(0, \\mathrm{e}]$时, $y=f(x)$的最小值为$3$, 求$a$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342072,7 +344263,10 @@ "id": "021434", "content": "设$a, b$是实数. 若函数$f(x)=a x^3-6 a x^2+b$, $x \\in[-1,2]$的最大值是$3$, 最小值是$-29$, 求实数$a$、$b$的值.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342091,7 +344285,10 @@ "id": "021435", "content": "设$a$是实数. 已知$a \\leq \\dfrac{1-x}x+\\ln x$对任意$x \\in[\\dfrac 12, 2]$恒成立, 求$a$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342110,7 +344307,10 @@ "id": "021436", "content": "设实数$t>0$, 函数$f(x)=t x^2+2 t^2 x+t-1$.\\\\\n(1) 对于$t$, 记$h(t)$是函数$y=f(x)$的最小值, 试用$t$表示$h(x)$;\\\\\n(2) 设$m$是实数. 若对于任意$t \\in(0,2)$, 关于$t$的不等式$h(t)<-2 t+m$恒成立, 求$m$的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342129,7 +344329,10 @@ "id": "021437", "content": "用长$24$米的钢条制作一个长方体框架(共$12$条棱). 已知长方体的长与宽之比为$2: 1$, 问: 该长方体的长、宽、高各为多少时, 其体积$V$最大? 最大体积是多少?", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342148,7 +344351,10 @@ "id": "021438", "content": "某公司生产一品牌产品, 已知生产总成本$C$(万元) 与销售量$x$(件) 满足函数关系式$C=2 x^3-x^2$, 销售收入总额$S$(万元)与销售量$x$(件) 满足函数关系式$S=15 x^2$, 为使该子公司利润$L$最大, 应当生产多少件该产品?", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342167,7 +344373,10 @@ "id": "021439", "content": "如图, 圆形纸片的圆心为$O$, 半径为$5 \\text{cm}$, 该纸片上的等边三角形$\\triangle ABC$的中心为$O$. 点$D, E, F$均为圆$O$上的点, $\\triangle DBC, \\triangle ECA, \\triangle FAB$分别是以$BC, CA, AB$为底边的等腰三角形. 沿虚线剪开后, 分别以$BC, CA, AB$为折痕折起$\\triangle DBC, \\triangle ECA, \\triangle FAB$, 使得$D, E, F$重合, 得到三棱锥. 求当$\\triangle ABC$的边长变化时, 所得三棱锥体积(单位: $\\text{cm}^3$)的最大值.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\filldraw (0,0) circle (0.03) node [above] {$O$} coordinate (O);\n\\draw (O) circle (2);\n\\draw (O) ++ (0:0.8) node [right] {$C$} coordinate (C);\n\\draw (O) ++ (120:0.8) node [above] {$A$} coordinate (A);\n\\draw (O) ++ (240:0.8) node [below] {$B$} coordinate (B);\n\\draw (O) ++ (60:2) node [above] {$E$} coordinate (E);\n\\draw (O) ++ (180:2) node [left] {$F$} coordinate (F);\n\\draw (O) ++ (300:2) node [below] {$D$} coordinate (D);\n\\draw (A) -- (B) -- (C) -- cycle;\n\\draw [dashed] (A) -- (E) -- (C) (C) -- (D) -- (B) (B) -- (F) -- (A);\n\\end{tikzpicture}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -342186,7 +344395,10 @@ "id": "021440", "content": "某农场有一块农田, 如图所示, 它的边界由圆$O$的一段圆弧$\\overset\\frown{MPN}$($P$为此圆弧的中点)和线段$MN$构成. 已知圆$O$的半径为$40$米, 点$P$到$MN$的距离为$50$米. 现规划在此农田上修建两个温室大棚, 大棚 I 内的地块形状为矩形$ABCD$, 大棚 II 内的地块形状为$\\triangle CDP$, 要求$A, B$均在线段$MN$上, $C, D$均在圆弧上. 设$OC$与$MN$所成的角为$\\theta$.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw (2,0) coordinate (R) arc (0:180:2) coordinate (L);\n\\draw (R) arc (0:{-asin(0.25)}:2) node [right] {$N$} coordinate (N);\n\\draw (L) arc (180:{180+asin(0.25)}:2) node [left] {$M$} coordinate (M);\n\\draw (0,0) node [below left] {$O$} coordinate (O);\n\\draw [dashed] (-2,0) -- (2,0) (0,2) node [above] {$P$} coordinate (P) -- ($(M)!(P)!(N)$);\n\\draw (40:2) node [above right] {$C$} coordinate (C);\n\\draw (140:2) node [above left] {$D$} coordinate (D);\n\\draw ($(M)!(C)!(N)$) node [below] {$B$} coordinate (B);\n\\draw ($(M)!(D)!(N)$) node [below] {$A$} coordinate (A);\n\\draw (M) -- (N);\n\\draw (A) -- (D) -- (C) -- (B);\n\\draw (C) -- (P) -- (D) (O) -- (C);\n\\end{tikzpicture}\n\\end{center}\n(1) 用$\\theta$分别表示矩形$ABCD$和$\\triangle CDP$的面积, 并确定$\\sin \\theta$的取值范围;\\\\\n(2) 若大棚 I 内种植甲种蔬菜, 大棚 II 内种植乙种蔬菜, 且甲、乙两种蔬菜的单位面积年产值之比为$4: 3$. 求当$\\theta$为何值时, 能使甲、乙两种蔬菜的年总产值最大.", "objs": [], - "tags": [], + "tags": [ + "第二单元", + "导数" + ], "genre": "解答题", "ans": "", "solution": "", @@ -356729,13 +358941,15 @@ "id": "030506", "content": "已知$\\triangle ABC$的三个顶点的坐标分别是点$A(5,1)$、$B(4,3)$与$C(0,-1)$, 直线$m:(k+2) x+(k-1) y+k-1=0$($k \\in \\mathbf{R}$).\\\\\n(1) 求直线$AB$的斜率与倾斜角;\\\\\n(2) 求边$AC$所在直线$l_1$的方程和边$AC$上的高所在直线$l_2$的方程;\\\\\n(3) 判断直线$m$与直线$AB$的位置关系;\\\\\n(4) 记$d$为点$A$到直线$m$的距离, 试问:$d$是否存在最大值? 若存在, 求出$d$的最大值; 若不存在, 说明理由;\\\\\n(5) 若直线$l$过点$C$, 且被两条平行直线$2 x+y-6=0$与$4 x+2 y-5=0$截得的线段长为$\\dfrac 72$, 求直线$l$的方程;\\\\\n(6) 若直线$l$过点$C$, 且点$A$与点$B$到直线$l$的距离相等, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题1", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356748,13 +358962,15 @@ "id": "030507", "content": "已知直线$l$过点$P(0,-1)$, 若直线$l$分别与直线$x+y-3=0$、$x-2 y+4=0$交于点$M$、$N$, 且点$P$恰为线段$MN$的中点, 求直线$l$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题2", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356767,13 +358983,15 @@ "id": "030508", "content": "(1) 若点$P(3,-2)$、$Q(-5,4)$到直线$l$的距离都等于$4$, 求直线$l$的方程;\\\\\n(2) 若点$P(3,-2)$、$Q(-5,4)$到直线$l$的距离都等于$\\lambda$, 试讨论: 对于不同的非负常数$\\lambda$, 满足条件的直线$l$有多少条?", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题3", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356786,13 +359004,15 @@ "id": "030509", "content": "若两定点$A$、$B$的距离为$3$, 动点$M$满足$|MA|=2|MB|$, 建立适当的平面直角坐标系, 求动点$M$的轨迹方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题4", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356805,13 +359025,15 @@ "id": "030510", "content": "椭圆$C: \\dfrac{x^2}4+\\dfrac{y^2}3=1$的左、右焦点分别是$F_1$、$F_2$, 点$T$是椭圆$C$上的动点.\\\\\n(1) 是否存在点$T$, 使得$\\angle F_1TF_2=90^{\\circ}$, 若存在, 求出点$T$的坐标; 若不存在, 请说明理由;\\\\\n(2) 求证: 当点$T$是椭圆短轴顶点时, $\\angle F_1TF_2$取得最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题5", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356824,13 +359046,15 @@ "id": "030511", "content": "已知双曲线$\\dfrac{x^2}{a^2}-\\dfrac{y^2}{b^2}=1$($a>0$, $b>0$)的左、右焦点分别是$F_1$、$F_2$, 在其渐近线上存在一点$P$, 满足$\\|PF_1|-|PF_2\\|=2 b$, 求该双曲线离心率的取值范围.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题6", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356843,13 +359067,15 @@ "id": "030512", "content": "抛物线的顶点在原点, 准线方程是$x=-1$, 是否存在被点$E(1,1)$平分的弦? 如果存在, 求出弦所在直线的方程以及弦长; 如果不存在, 请说明理由.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题7", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356862,13 +359088,15 @@ "id": "030513", "content": "过抛物线$C: y^2=4 x$的焦点$F$的直线交$C$于$A$、$B$两点, 过$A$、$B$两点分别作$C$的准线的垂线, 垂足为$A_1$、$B_1$, 以线段$A_1B_1$为直径的圆$E$过点$M(-2,3)$, 求圆$E$的方程.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题8", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356881,13 +359109,15 @@ "id": "030514", "content": "已知两圆$C_1:(x-2)^2+y^2=54, C_2:(x+2)^2+y^2=6$, 动圆$M$在圆$C_1$内部且和圆$C_1$内切、和圆$C_2$外切.\\\\\n(1) 求动圆圆心$M$的轨迹$C$的方程;\\\\\n(2) 过点$A(3,0)$的直线与(1)中的曲线$C$交于$P$、$Q$两点, 点$P$关于$x$轴对称的点为$R$, 求$\\triangle ARQ$面积的最大值.", "objs": [], - "tags": [], + "tags": [ + "第七单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修第一册章复习课例题与习题试题9", + "origin": "空中课堂选择性必修第一册章复习课例题与习题", "edit": [ "20221229\t王伟叶" ], @@ -356900,13 +359130,16 @@ "id": "030515", "content": "已知随机变量$X$服从正态分布$N(\\mu, \\sigma^2)$($\\sigma>0$), 且$P(X \\leq c)=P(X>c)$, 则$c$的值为\\bracket{20}.\n\\fourch{$0$}{$\\sigma$}{$-\\mu$}{$\\mu$}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "正态分布" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第七章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -356919,13 +359152,16 @@ "id": "030516", "content": "设两个正态分布$N(\\mu_1, \\sigma_1^2)$($\\sigma_1>0$)和$N(\\mu_2, \\sigma_2^2)$($\\sigma_2>0$)的正态密度函数图像如图所示, 则\\bracket{20}.\n\\begin{center}\n\\begin{tikzpicture}[>=latex]\n\\draw [->] (-4,0) -- (4,0) node [below] {$x$};\n\\draw [->] (0,0) -- (0,2.5) node [left] {$y$};\n\\draw (0,0) node [below left] {$O$};\n\\draw [domain = -4:4,samples = 300] plot (\\x,{1/0.2/sqrt(2)/sqrt(pi)*exp(-pow(\\x+1,2)/2/0.2/0.2)});\n\\draw (-1,2) node [above] {$N(\\mu_1,\\sigma_1^2)$};\n\\draw [domain = -4:4,samples = 300] plot (\\x,{1/0.5/sqrt(2)/sqrt(pi)*exp(-pow(\\x-0.5,2)/2/0.5/0.5)});\n\\draw (0.5,1) node [above,fill = white] {$N(\\mu_2,\\sigma_2^2)$};\n\\end{tikzpicture}\n\\end{center}\n\\fourch{$\\mu_1<\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1<\\mu_2$, $\\sigma_1>\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1<\\sigma_2$}{$\\mu_1>\\mu_2$, $\\sigma_1>\\sigma_2$}", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "正态分布" + ], "genre": "选择题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第七章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -356938,13 +359174,16 @@ "id": "030517", "content": "已知随机变量$X$服从正态分布$N(\\mu, \\sigma^2)$($\\sigma>0$), 求$P(\\mu \\leq X<\\mu+\\sigma)$的值. (结果精确到$0.1 \\%$)", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "正态分布" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第七章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -356957,13 +359196,16 @@ "id": "030518", "content": "已知某批零件的长度误差(单位: 毫米)服从正态分布$N(0,3^2)$, 从中随机取一件, 求长度误差落在区间$(3,6)$内的概率. (结果精确到$0.1 \\%$)", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "正态分布" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第七章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -356976,13 +359218,16 @@ "id": "030519", "content": "已知随机变量$X$服从正态分布$N(2, \\sigma^2)$($\\sigma>0$), 且$P(X<4)=a$, 求$P(X<0)$. (用$a$表示)", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "正态分布" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第七章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -356995,13 +359240,16 @@ "id": "030520", "content": "某批待出口的水果罐头每罐净重$X$(单位: $\\text{g}$)服从正态分布$N(184,2.5^2)$, 求:\\\\\n(1) 随机抽取一罐, 其净重不小于$191.5 \\text{g}$的概率;\\\\\n(2) 随机抽取一罐, 其净重在$179 \\text{g}$与$189 \\text{g}$之间的概率.(结果精确到$0.1 \\%$)", "objs": [], - "tags": [], + "tags": [ + "第八单元", + "正态分布" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第七章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357014,13 +359262,15 @@ "id": "030521", "content": "下表给出了一些地区鸟的种类数与该地区的海拔高度的数据, 试绘制散点图, 并通过观察散点图大致判断鸟的种类数与海拔高度之间的相关性.\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline 地区 & A & B & C & D & E & F & G & H & I & J & K \\\\\n\\hline 海拔高度 /m & 1250 & 1158 & 1067 & 457 & 701 & 731 & 610 & 670 & 1493 & 762 & 549 \\\\\n\\hline 鸟的种类 / 种 & 36 & 30 & 37 & 11 & 11 & 13 & 17 & 13 & 29 & 4 & 15 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357033,13 +359283,15 @@ "id": "030522", "content": "随机抽取某地区的 7 家超市, 得到其广告支出与销售额数据如下:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline 超市 & A & B & C & D & E & F & G \\\\\n\\hline 广告支出/万 & 1 & 2 & 4 & 6 & 10 & 14 & 20 \\\\\n\\hline 销售额 /万元 & 19 & 32 & 44 & 40 & 52 & 53 & 54 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n请推断该地区超市的销售额与广告支出之间是否存在线性相关关系.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357052,13 +359304,15 @@ "id": "030523", "content": "随机抽取 10 家航空公司, 对其最近一年的航班正点率顾客投诉次数进行调查, 所得数据如下:\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}\n\\hline 航空公司编 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline 航班正点率/\\% &$81.8$&$76.8$&$76.6$&$75.7$&$73.8$&$72.2$&$71.2$&$70.8$&$91.4$&$68.5$\\\\\n\\hline 顾客投诉/次 & 21 & 58 & 85 & 68 & 74 & 93 & 72 & 122 & 18 & 125 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n试判断航班正点率与顾客投诉次数之间是否存在线性相关关系.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357071,13 +359325,15 @@ "id": "030524", "content": "某地区的环境条件适合天鹅栖息繁衍. 有人发现了一个有趣的现象, 该地区有 5 个村庄, 其中 3 个村庄附近栖息的天鹅较多, 婴儿出生率也较高; 2 个村庄附近栖息的天鹅较少, 婴儿出生率也较低. 有人认为婴儿出生率和天㧴数之间存在相关关系, 并得出一个结论: 天鹅能够带来孩子. 你同意这个结论吗?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357090,13 +359346,15 @@ "id": "030525", "content": "有人收集了某城市居民年收入(所有居民在一年内收入的总和)与$A$商品销售额的$10$年数据, 如下表所示:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\n第$n$年 &$1$&$2$&$3$&$4$&$5$&$6$&$7$&$8$&$9$&$10$\\\\ \\hline\n居民收入/亿元 &$32.2$&$31.1$&$32.9$&$35.8$&$37.1$&$38.0$&$39.0$&$43.0$&$44.6$&$46.0$\\\\ \\hline\n$A$商品销售额/万元 &$25.0$&$30.0$&$34.0$&$37.0$&$39.0$&$41.0$&$42.0$&$44.0$&$48.0$&$51.0$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n请画出散点图, 观察成对数据是否线性相关, 并通过计算相关系数判断``居民年收入'' 和``$A$商品销售额''的相关程度.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357109,13 +359367,15 @@ "id": "030526", "content": "对下面两组数据\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline $x$ & 1 & 2 & 3 & 4 & 10 & 10 \\\\\n\\hline $y$ & 1 & 3 & 3 & 5 & 1 & 11 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n计算相关系数, 大概在$0.5$左右. 通过观察散点图, 发现对这两组大部分数据来说, 变量$x$与$y$有很强的线性相关关系, 是什么因素导致相关系数只存$0.5$左右?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357128,13 +359388,15 @@ "id": "030527", "content": "已知根据某样本数据可得到回归方程为$y=4 x+\\hat{b}$, 且$\\overline x=3$, $\\overline y=6$, 则$\\hat{b}=$\\blank{50}.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357147,13 +359409,15 @@ "id": "030528", "content": "某小吃店的日盈利$y$(单位: 百元)与当天平均气温$x$(单位:$^{\\circ} \\text{C}$) 之间有如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline $x/^{\\circ} \\text{C}$&$-2$&$-1$& 0 & 1 & 2 \\\\\n\\hline $y$/百元 & 5 & 4 & 2 & 2 & 1 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n甲、乙、丙$3$位同学对上述数据进行了分析, 发现$y$与$x$之间具有线性相关关系, 他们通过计算分别得到$3$个线性回归方程:\\\\\n\\textcircled{1} $y=-x+2.8$; \\textcircled{2} $y=-x+3$; \\textcircled{3} $y=-1.2 x+2.6$. 使得拟合误差$Q$取到最小值的方程是\\blank{50}(填序号).", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "填空题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357166,13 +359430,15 @@ "id": "030529", "content": "气象部门由每天的最高气温的数据, 得到每月最高气温的平均数, 简称平均高温.表格是$2017$年$31$个城市$1$月和$7$月的平均高温数据(单位: $^\\circ\\text{C}$).\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\hline\n城市 & 1月 & 7月 & 城市 & 1月& 7月& 城市 & 1月& 7月\\\\ \\hline\n北京 &$3$&$32$& 南京 &$9$&$35$& 成都 &$12$&$32$\\\\ \\hline\n南宁 &$20$&$33$& 重庆 &$12$&$36$& 上海 &$10$&$36$\\\\ \\hline\n福州 &$17$&$36$& 沈阳 &$-3$&$31$& 广州 &$21$&$33$\\\\ \\hline\n石家庄 &$3$&$33$& 贵阳 &$9$&$28$& 太原 &$3$&$32$\\\\ \\hline\n哈尔滨 &$-11$&$30$& 天津 &$3$&$33$& 海口 &$22$&$32$\\\\ \\hline\n乌鲁木齐 &$-3$&$32$& 杭州 &$11$&$36$& 武汉 &$10$&$34$\\\\ \\hline\n合肥 &$9$&$35$& 西安 &$8$&$36$& 呼和浩特 &$-3$&$30$\\\\ \\hline\n西宁 &$4$&$27$& 济南 &$6$&$33$& 银川 &$2$&$32$\\\\ \\hline\n昆明 &$17$&$24$& 长春 &$-8$&$29$& 拉萨 &$8$&$23$\\\\ \\hline\n长沙 &$11$&$35$& 兰州 &$5$&$33$& 郑州 &$7$&$34$\\\\ \\hline\n南昌 &$13$&$35$& & & & & & \\\\ \\hline\n\\end{tabular}\n\\end{center}\n(1)画出并观察各城市 1 月与 7 月的平均高温的散点图, 你认为 1 月平均高温$(x)$与 7 月的平均高温$(y)$有线性关系吅? 描述散点图的特点;\\\\\n(2) 结合地理知识并用统计方法分析表中的数据, 解释这两个月平均高温的关系.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357185,13 +359451,15 @@ "id": "030530", "content": "为了解小麦种子是否灭菌与小麦发生黑穗病的关系, 经试验观察, 得到如下数据. 根据这组数据, 能否认为发生黑穗病与种子是否灭菌有关?\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline & 种子灭菌 & 种子未灭菌 & 总计 \\\\\n\\hline 有黑穗病 & 26 & 184 & 210 \\\\\n\\hline 无黑穗病 & 50 & 200 & 250 \\\\\n\\hline 总计 & 76 & 384 & 460 \\\\\n\\hline\n\\end{tabular}\n\\end{center}", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357204,13 +359472,15 @@ "id": "030531", "content": "一医疗团队为研究某地的一种地方性疾病与当地居民的卫生习惯 (卫生习惯分为良好与不够良好两类) 的关系, 在已患该疾病的病例中随机调查了 100 例 (称为病例组), 同时在未患该疾病的人群中随机调查了 100 人 (称为对照组), 得到如下数据:\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& 不够良好 & 良好 \\\\ \\hline\n 病例组 & $40$&$60$\\\\ \\hline\n 对照组 & $10$&$90$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n试问: 能否认为患该疾病群体与未患该疾病群体的卫生习惯有差异?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357223,13 +359493,15 @@ "id": "030532", "content": "一个随机抽样的样本包括 110 位女士和 90 位男士, 女士中约有$9 \\%$是左利手 (又称左撇子), 男士中约有$11 \\%$是左利手. 基于这些数据, 你认为左利手与性别有关吗? 为什么?", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ], @@ -357242,13 +359514,15 @@ "id": "030533", "content": "某儿童医院用甲、乙两种疗法治疗小儿消化不良. 采用简单随机抽样的方法对治疗情况进行检查, 得到了如下数据: 抽到接受甲种疗法的患儿$67$名, 其中未治愈$15$ 名, 治愈$52$名; 抽到接受乙种疗法的患儿$69$名, 其中未治愈$6$名, 治愈$63$名. 试分析甲、乙两种疗法的效果是否有差异.", "objs": [], - "tags": [], + "tags": [ + "第九单元" + ], "genre": "解答题", "ans": "", "solution": "", "duration": -1, "usages": [], - "origin": "空中课堂选择性必修分布与成对数据补充习题", + "origin": "空中课堂选择性必修第八章补充习题", "edit": [ "20230101\t王伟叶" ],